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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I ■ ■ ^^H THE HE'-v VOHK 1 PUBLIC UBHAKY M AstOF, Lrwr^X H 1 J / Lessons in Horology BY Jules Grossmann Director of the Horological School, of I/Kle, Switzerland AND Hermann Grossmann Director of the Horological and Electro-Mechanical School, of Neuchatcl, Switaerland AUTHORIZED TRANSLATION By JAMES ALLAN, JR. of Charleston* S. C. Former Pupil of the Horological School, of I#ocle, Switzerland VOLUME I The Principles of Cosmography and Mechanics Relat- ing to the Measurement of Time — Motive Force, Mainsprings, Trains, Gearings, etc. IVITH OVER too ILLUSTRATIONS PUBLISHED BY THE KEYSTONE THE ORGAN OF THS JSWSI*RY AND OPTICAI, TRADES 19TH & Brown Sts., Philadelphia, U.S.A. 1905 . ■ **■ • • . All Rights Reserved , . - ' M ji'^ HHV ] IDLES GKOSSMANK -BsburualEbfrB- J ivaWe, Desr Berlin, Uermnny, oi. July ^ 28, 1329. Ue tegan hi. horologi.al garter in his nativB town when fllleen yesra ,.l.i, soon moving to Berlin. He snl«e- quontiy -orked and studied in the Brilish A I.le> and P»ri>, linally setlllos in Loclr, ^1l Switzerland, where hii great wlenllBe tri- umph. »ero aehieved. He has been muuh in Locie, of whleh he became IXreelor. Hia .ebiBveraentB in Ihe field of horology hive bpeu Impoilsiit factors in the advancenient lln^moit profonnd knowledge of the BulJeoL ll waa at Ihe instancB of ihe 8«iBa Guvem- IrealJM "Leasons lu llorolo((y," which ii J0LE8 GEcaSlLiSS 1 T KHM A.VN GUOSSMAKS, son ofJiilr-., |-1 „« ^rn i. L«lo. 8«l,..rLaud. .n April ^ I3S3. At Ihp ige of liitnn ul thB Horuidglial School of T.oele. nuder to Switierlaud to ulUl further [.urBmi bia jL^ m Biudies in Uigber boralogj. When uuly th» poaitiDii of Director of tlie Ilorol.igtcal ■lid ElH<ro-Mei.'haalGiil Schoul, of Nuu- uhnlel, Swil«rl.nd. The wort of this ^y^^J HEhool 9000 became fmiiouH. beiog iwsrded rannj honon Bt the grent InH-rofllinoaJ ti- linnlioiis were oonfelred od the Director. " Ifflwoni in HoroloET," which complelely Onvnnr tlut Bilh!.-!*! In Ihanvir nnH nnifJEiM- '-■"'■■ 1 w^ 5 .-. i-4 . !'')R01.0GY .r.^tr.'t:**^ '.I I < . < ■ r i « ■ ■> ' \ 39840A • « «• GOPTBIGHT, 1905, BT B. THOBPB Publisher of Thb Ketstons • • * » • • • • • f 9 • \ t PREFACE BY THE AUTHORS ■^TO one ignores the fact tliat horology has attained a highly prominent position among the mechanical arts during the past quarter of a century, anci that this lact is due to the rapid progress of science, industry and commerce in our epoch. If it sufficed formerly to know the time, approximately, and to make use of the indications of the sun dials, town clocks with single hands and other primitive instruments, modern times, with their multi- plied requirements, have rapidly despoiled us of this ancient simphcily; they demand of us an exact marking of every instant, 'hich only modern horary instruments permit us to attain. The magnificent chronometers, whose superior timekeeping re admire to-day, are the product of Uie two-told effort; first, of the theorist who, by his calculations, determines all the principles; then of the practical workman who faithfully follows these in the execution of his work. In the period in which we live we cannot believe that we have yet attained the highest point of precision, but the results are already so brilliant that the mind now asks the question whether, before going farther into the technical domain, it would not be better first to bring to perfection the means of observation and of rating, which we now invariably employ with a degree of uncertainty. The magnificent instruments to which we have alluded are, more- over, still exceptions; they are very expensive as yet. So the most practical object of the technical study of horology is to approach, as nearly as possible in public timepieces, the results of the precision chronometer, at least so far as concerns the exactness of timekeeping. This purpose will surely be attained when horology, seconded by the admirable resources of mechanics, will entirely cease to be an art, too often empirical, and become a purely mechanical science. It has long been believed that the theory of horology formed a science by itself, independent of general mechanics, and for a long time the watchmaker would not listen to anything about mechanics, I vi Preface pretending that it was impossible to apply its data to the minute pieces which compose the mechanism of a pocket watch. This assertion was often apparendy sustained by practical results, and frequendy the purely mechanical data appeared as if they could not be applied to horology. But this conclusion, let us hasten to say, was false; for the reason that the mathematical formulas employed in mechanics often require less development than when they are applied to horology. In the first case many of the terms could be neglected which in the second would become important. We must not be astonished, either, if the results are not always what we seek. Let lis take an example: Would one really dare to pretend that the laws of friction established by Coulomb, are inexact because it is very difficult in horology to separate friction proper from the influence of adhesion produced by the oil or other lubricating material ? This second factor, which may often be omitted in large mechanics, becomes, we know, an important factor in horology. The work which we present to the English-speaking watchmakers is written by watchmakers and for watchmakers, and with the idea that horology and mechanics are twin sisters, and that the same laws and the same rules control both. This work is the fruit of long e-xperience in the domain of professional instruction in horological schools. We have endeavored to avoid speculadons purely theoretical, as well as long descriptive explanations, which belong to books suited to the general public. If the solution of some problems cannot, in our estimation, be accomplished without the aid of higher mathematics, because of the precision required to attain the desired end, it must be noted that these questions can generally be put aside by those persons to whom the subtleties oi mathematical analysis are unfamiliar. It is sufficient then to recognize the fact that the calculations have been made to verifj' the deductions and to make use of the conclusions which may be drawn from them. We are also obliged to grade the difficulties of calculation so that they are presented in proportion to the development of the mathematical knowledge of the persons who undertake the study, and we follow each problem with at least one numerical application. When it is possible we give also together with a complicated solution, another similar to it, but more simple. plan comprises, first, a short introduction on the principles of cosmography and mechanics having relation to the measuring of Preface vii time. Then follow chapters which are devoted to the study of motive forces produced by the weight and the barrel spring, the calculations of trains and the theory of gearings. Then chapters on escapements, and finally the theory of adjusting and regulating forms an important part of the work, and will be treated with all the exactness due the subject. We will close this exposition of the theory by a treatise on the compensation of chronometers. We hope that this work will contribute its share towards forming a generation of capable and educated horologists who can assist in the development of the fascinating industry of horology. We owe a just tribute of appreciation to The Keystone, which has undertaken the publication of this work in the English language^ and to James Allan, Jr. , of Charleston, S. C. , former pupil of the Lode Horological School, who has so well performed the work of translation. Jules Grossmann, Lode, Switaseiland. Hermann Grossmann, Neachate) Switzerland. ^H 9 ^K 19 ^H aS ^B 39 H ^ H 37 H ^ ^m 39 H 43 TABLE OF CONTENTS. enphL p 1 ■ 3 19 ao 33 33 23 H *1 34 35 36 a? ■3a INTRODUCTION. I. General Principles of Cosmography Relating to Horology. Determiiiatioa of the position of a point on the terrestrial sphere II. General Principles of Mechanics. Definition Work of a force tangent to a wheel CHAPTER I. General Functions of Clocks and Watches. ti) Lessons in horology. Wheel-work. — lb purpoie in the mechaniim of dock) and watcha . Etcftpemunti CHAPTER II. Maintaining or Motive Forces. The wei^t m a motive force . , . . The bMTel ipring u & motive force a spring 43 Measurement ol the force of a Thear«ticat itudy oE the momeni of a ipring** farce . Coefficient of elasticity Variation of tfie coefficient of elasticity 47 Values of the coefficient of elasticity E Limit of elasticity , Moment of the elastic force of a spring subjected to a flexion . Inequality of the elastic force of the spring 56 Length of the spring 5S Development of a ipring 39 Diameter of the hub 64 Work produced by ipring 6S The fuiee .... Calculation of the variable raiJius of the fusee's helix 6g . Uniformity of the force of the spring in fusee watclies 74 Stop-work 78 Geometrical construction of the Maltese cross stop-work : CHAPTER IIL Wheel-Work. 1J9 Purposes of wheel-work , Calculationj of train) 130 Calculations of the number of turns 133 Calculation of the number of oscillations of the balance 84 136 Calculations of the numbers of teeth 86 Problenii relative to the preceding qimtiaiu 8g Table of Contents. 3 141-145 Numbers of turns B?-^ 146 Numbers of oscillations of the balance 90 147-149 Numbers of teeth 91^93 ISO Numbers of leeth of the minute wheels. Description of this iM-'H Numbers of teeth of an astronomical clock 97-99 'SS-'sS Numbers of teeth of lost mobiles loo-ioa '59 Indicator of the development of the spring in fusee timepieces . . loi i&> Simple calendar watches 104 '61 Decitnal watches 106 164 Calculation of numbers comprising the teeth-ranges of the wheels of a watch with independent second hand 109 iS8-i59 Wheel-work of the stem-winding mechanism 118-113 i^a Calculation of the train in a watch of the Roskopt type .... 114 » CHAPTER IV. fl Gearings. H DeGnition 117 I Practical examination of a gearing iiS I Flnt — IHflance of the c«nten iig H '7& Primitive radii tig I '79 Applications lai I 1S5 Calculation of the primitive radii 125 ' I93 Application of the theory of primitive radii to the escajiements . . 129 Second — Shape of the teeth and leavei 130 J95 General study of the transmission of force in gearings 130 OeterminalioD of the fonni of contact in geariod* 1 36 io9 First — Graphic method. Exterior gearing 137 SID Interior gearing 138 sii Second — Method of the envelopes 140 aig Evolvents of circle gearings 146 MO Third— Roller method 148 121 Flank gearings 150 136 a Determination of the profile of a tooth corresponding to a profile chosen arbitrarily 154 aa6i Gearings by the evolvent of a circle 155 Teeth-rftnge 156 ^^ Third—Total ditunet«n IS9 4 Lessons in Horology. Cycloid - 13 J3S Definition 15] 236 Drawing oi the cycloid 15 337 Drawing of the cycloid of a continuous movement i6( 138 Normal and tangent to the cycloid j6! 239 Evolute and radius of curvature oi the cycloid 16] 140 Length of the cycloid ifil Cpicyclrad l£{; 141 Definition 16^ lit Drawing of the epicycloid i^ 243 Drawing of the epicycloid of a continuous movement i& J44 To draw a normal, then a tanget to the epicycloid ifi 945 Evolute and radius of curvature of the epicycloid 161 350 Length of the epicycloid i6| »5i-2S3 Applications 170-17: RelatioD of the radtui vector to the angle formed bjrtbe VBiukbl* raiUiii vector and the initieJ radim vector Table showing the angle traversed by the pinion of several ordi- nary gearings during the contact of a tooth of the wheel with the leaf oi this pinion r7j Calculation of the total raditn of the wheel 176 Form of the exceu of the pinion letf in a flaak ^earin^ 176 a68 Radius of curvature of an elipse 270 Total radius of the pinion GraphicBl conitniction ef ge&ringi PractiCB) Bpplicstioni of the theory of gearingi 276 The proportional compass and its use Tabie for using the proportional compass 195 384 Verification of a proportional compass 196 385 Determination of the distance of the centers of a gearing by means of the proportional compass and of a depthing tool . . 196 286 The proportional compass and stem-winding wheel gearings . . . 197 286 Gearing of the crown wheel in the ratchet wheel 198 287 Gearing of the winding pinion in the crown teeth of the contrate wheel 199 288 Gearing of the sliding pinion and of the small settiug wheel . . . 200 289 Gearings of the dial wheels aoi Vaiioui calculationt relative to gearing 203 Conical geariogi 214 315 Form of the teeth ai6 Table of Contents, 5 Fuagnphg. 318 Construction of conical gearings 218 Defects which present themselves in these gearing 223 Passive resistances in gearing 227 326 General ideas 227 Friction 228 330 The two kinds of friction 229 332 Laws of friction 229 333 Experimental determination of the force of friction 230 335 Table of the coefficients of friction 231 336 Work of friction 232 337 Angle of friction 233 Calculation of the friction in ^earin^s 234 339 Friction of the teeth ranges 234 346 Friction before and after the line of centers 238 348 1st. The wheel drives the pinion after the line of centers .... 239 349 2d. The pinion drives the wheel before the line of centers ... 241 350 3d. The wheel drives the pinion before the line of centers . . . 242 351 4th. The pinion drives the wheel after the line of centers .... 244 352 Recapitulation of the preceding calculations 244 Calculations of the friction of pivots 245 354 Work absorbed by the friction of the plane surface of the shoulder of a pivot 245 356 Work absorbed by the friction of the cylindrical surface of a pivot 247 357 Determination of the lateral pressure received by the pivots of the mobiles in a train 247 Influence of the oil 253 Application of the theory of gearings 255 Functions of the heart in chronographs 255 LESSONS IN HOROLOGY LESSONS IN HOROLOGY. y COURSE IN MECHANICS AS APPLIED TO CHRONOMETRY. I. Introduction —General Priaopl«i of Cotmogntphj' Relating to Horoto^. 1. Principles of the measurement of time. Cosmography is » science which has for its object the study of the different celestial phenomena as they are given to us by observation and calculation ; it comprehends also the study of the principles which are the basis ^^pf the measuring of time. ^^L When one proposes to measure a length, surface, volume or ^^Height of any kind one chooses arbitrarily a unit of length, surface, ^^Holume or weight with which one compares the object to be ^^Eneasured, noting exactly the number of times that this unit is ^^Hontained therein. ^^B When the purpose is to measure the intervals of time, it is no ^^^bnger possible to make use of an analogical method. But to ^^^Hect this operation one is obliged to determine the space traversed ^^Ey a body animated with a uniform or periodically uniform motion (31). In the first place one concludes that the intervals of time are proportionate to the spaces traversed by the body considered. 2. It is necessary then to admit that all measurements of time must be deduced from the observation ol a regular movemenL Thus, formerly, one determined the fraction of timCj more or less great, by the running of sand in the "hour glass," or of water in the "clepsydra." Now time is measured in docks and in watches by the periodically uniform movement of the pendulum or balance wheel. 3. Units of time. Sidereal day. Solar day. For the determi- nation of the unit of time it is necessary to choose the most unifonn movement possible, a movement whose speed must be the same to-day, to-morrow, in a year or in an indefinitely prolonged period. Such a movement filling this condition absolutely, is the rotation of the earth on its axis ; no cause or effect whatever could increase or diminish it. We have positive proofs that this movement is the same to-day as it was in the time of Hipparchus, an ancient astronomer of the school of Alexandria who lived two centuries lo Lessons in Horology. before Jesus Christ. We can assure ourselves by the calculation of the eclipses, that the length of one of these rotations is the same to-day as in the time ol that astronomer within jj-j of a second. This movement has then been chosen because of its great regu- larity as the basis for the measuring of time. The duration oi a complete rotation is the unit, and is called 4. In order to determine with exactitude the commencement and the end of this movement, it is necessary to choose a point of repose outside ol the earth, and for this purpose a fixed star or the sun is taken. Let us remark that the result differs according as we take one or the other of these two points. The following demons- tration will explain the reason. 5. We know that the earth not only turns on its axis, but that it has also a simultaneous movement around the sun. Let us take, then T and T' . Fig. i, as the two positions which the earth occupies in its orbit at the commencement and at the end of one of its diurnal rotations. In the first of these positions, a is a point on ita Hurface Irom which can be seen at this instant the center of the ■uii S, in an imaginary plane passing through the two poles and the point considered a, this plane is the meridian plane. At the end of a certain time, the earth has traveled in its orbit to the position T* and the point a arrives at a' in such a manner that the line 7* a' is parallel to T a. The earth wHl then have accomplished one rotation on its axis and all of its parts will have, with relation to the fixed stars, the same positions that they had at T. The time during which this rotation is accomplished is called^ H lidereal day, But from the point a' in the position T' one could not see the •Un in the meridian plane ; in order that the observer placed at t£ could perceive it anew in this plane, it would be necessary for the pi)lnt «' to be removed to b in traversing the arc a' b. The solar ttt^y, thut is to Biiy, the time which elapses between two consecutive. IWHHUgeN (i( the nun to the meridian plane is then longer than the ■IflvrcHl (tiiy. In ilividing the solar day into 24 hours, the hour Intrt ficj iTihiiit<-N imd the minutes into 60 seconds, the sidereal day uniiiitH only -J3 h(jun<, ,<^6 minutes, 4.09 seconds ; the sidereal day is Iht^ii ■Imrtrr thun the hoW day 3 minutes. 55.91 seconds. It we divide, on the other hand, the sidereal day into 24 hours, Ih" ttulm diiv will count 34 hours, 3 minutes, 56,55 seconds. Thia General Principles of Cosmography, II value of the solar day, variable from the sidereal day, is only a mean value. 6. The time that the earth takes to traverse its orbit, that is to say a year, contains exactly one sidereal day more than the solar days. 7. True time. Mean time. The curve that the earth describes Fig. 1 around the sun is an ellipse of which that star occupies one of the foci. Our planet does not traverse this curve with a uniform speed, it moves more rapidly when it is nearest the sun and more slowly when it is farthest away. The arcs traversed by the earth in one day are not then the same length during all the year. There results an irregularity in the duration of the solar day, the solar day is longest when the earth goes fastest, and it is shortest when Lessons in Horology. its movement of translation is slowest. Another cause which still increases this irregularity is due to the fact that the axis of the earth is not perpendicular to the plane of the orbit that it traverses around the sun (plane of the ecliptic). The length of the solar day can vary in 34. hours as much as 30 seconds, plus or minus. Thus the solar day with its diurnal variation of length does not fill at all the conditions desired for the measuring of time, the unit adopted must be of fixed value, so that our horological instruments, all based on a uniform movement, may follow their regular running without necessitating perpetual resetting. We fall then into a difficulty, since naturally the sun should measure the time for us, while in reality its unequal move- ment does not lend itself to this measuring. The difficulty has been adjusted as follows. We divide the total duration of the year by the number of solar days that it contains ; the quotient will be a mean value, shorter than the solar days of greatest length and longer than the solar days of least duration. It will be, moreover, almost equal to certain days between them. This mean value is called ?nean lime. We call, on the contrary, Inie time the direct interval of time elapsing between two successive passages of the sun across the meridian. The difference, plus or minus, between true time and mean time can amount to as much as 17 minutes. 8. The equation of time is the value that must be added to or subtracted from the true solar day to obtain the mean solar day. The year book of the Bureau of Longitudes announces each year in a calendar the result of the equation of time, and gives in a column entitled "Mean time at true noon" what a chronometer, regulated on mean time, should indicate at the exact moment of noon. The equation of time is nothing or almost that, four times a year — the 15th of April, 15th of June, 31st of August and 25th of December, while it attains its greatest value between the loth and i2th of February and the first days of November. 9. Laylnff out of a meridian line. We already have an idea of the importance of the meridian plane in the determination of the length o! a rotation of the earth on its axis. Let us see now how we can proceed to establish the direction of a meridian line, that is to say, of the trace of the meridian plane on the surface of the earth. Among the several methods known let us choose the following, which recommends itself on account of its ex- treme simplicity, and which does not require instruments of precision. General Principles of Cosmography. 13 On a horizontal plane, conveniently placed, we fix a vertical style ; from its toot, O, Fig, 2, we describe on this plane several concentric circles of any size, such as m n, m'n', etc Let us mark ton these circumferences the points A, B, C, etc. , where the extremity of the shadow of the pin reaches Ijefore midday. In the afternoon renew the operation by indicating in the same manner the points , if, C, etc. We connect the points marked on the same cir- cumference by a straight line, and we obtain thus as many straight lines as circles, and they are parallel to each other. The perpen- dicular laid o5 from the center O, on these straight lines, will be the meridian line sought. Since the shadow cast by such a style is never very distinct, we will reach a greater accuracy by finishing the extremity oi the pin with a metallic plate, in which we pierce a fine hole. We indicate then the center of the image of the sun on each of the circumferences, as previously done. In the above con- struction a single arc sufficed, but it is preferable to employ several hich mutually control each other, the middle of the straight lines, 'A', BB', CC, etc., should be found with the center O on the le perpendicular N. 10. When the middle of the small image of the sun is found on the meridian it is nearest the foot of the style and consequendv the sun is at its greatest height ; it is then exactly midday. To obtain the mean hour we must consult a table ol equations and add or 14 Lessons hi Horology. subtract, according to the season of the year, the correction indi- cated for this day. Generally, as we have said, these tables indicate the "mean time at noon," thus for the 17th of November, 1893, we find for example indicated : Mean time at noon 1 1 h. 45 m. 10 s. the clock or watch should then mark 11 hours, 45 minutes and 10 seconds when the middle of the small image of the sun is projected on the meridian line, U. The Meridian Glass employed in the observatories is nothing else than a meridian line determined with the greatest exactness. It is generally a glass of sufficiently large size which can only move, in the meridian plane, and is divided by it into two symmetrical parts. It is supported on two immovable pillars by means of trun- nions, which permit it to take all the positions possible around its axis of rotation. It can then be used to observe the passage on the meridian of all the visible stars above the horizon. The sensii ness of the instrument is moreover augmented by the magnifying power of the glasses employed. Since such an instrument cannot be transported, we have recourse to other instruments in order to determine the hour at any locality, on the sea for example. The one generally used is the sextant, but its employment is complicated. 12. If the mean hour is known, and it is only a question of maintaining it, the apparent motion of the fixed stars is easily used for this purpose ; in short, since these stars return to the same position at the end of 34 sidereal hours, it is sufficient to place j level of any sort, but fixed and invariable, in the direction of a star the next day at the same hour (solar time), less 3 minutes, 55.9: seconds this same star would present itself anew before the level The fixed stars afford the greatest facility for the control of the running of watches and clocks. 13. Determination of the position of a point on the terrestrial sphere. Since marine chronometers are among the instruments wiiich are used to determine the position of any point on the surface of the earth, especially that of a vessel at sea, each watchmaker should inform himself of the part that these instruments play in ■uch observations, on which depend the security of the ship and that of the beings which it transports. 14. In ordtT to represent the position of a point on the surface (if a sphere, such as the earth, we suppose described on this sphere two great circles, one, passing through the two poles, General Principles of Cosmography. 15 called the meridian circle; the Other, perpendicular to the first, is the equator. This last is consequently perpendicular to the terrestrial axis, and at all points equally distant from the poles. Each of these circles is divided into 360 degrees. The divisions marked on the meridian circle commence at the equator, and are reckoned north and south to the poles, therefore from o to 90 degrees. These degrees are called degrees of latitude. Since we can imagine an infinite number of meridians passing through the poles, the point o can be placed arbitrarily, that is to say, the first meridian at whatever place it suits best. Thus England has chosen as the starting point the meridian which passes through the Observatory at Greenwich, in the neighborhood of London ; France has made choice of the one which passes through the Observatory of Paris, and other nations have made their first meridian pass through the Isle of Fer. The degrees reckoned on the equator are called degrees of longitude, and are reckoned both to the east and to the west of the first meridian, from o to 180 degrees. By imagining circles parallel to the equator passing through each division of the meridian circle, and meridian circles passing through each division of the circle of the equator, the latitude of a point will be then the distance in degrees from the parallel circle passing through this place to the equator, and its longitude will be the distance in degress from the meridian of this place to the idian chosen as the starting point. These values constitute what are called the geographical '<rdinafes of a point, and the position of this point on the terres- globe will be perfectly determined when we know its longitude east or west, and its latitude north or south. Thus we would say that the geographical coordinates of the city of Neuchatel, in Switzerland, are ^H 4G° 59' 15" north latitude, ^p 4° 35' 54" longitude, east of the meridian of Paris. 15. In order to determine practically the latitude of a point A, Fig. 3, the simplest manner is to measure the angle formed by a horizontal line A B, and by the line A C ending in the Polar star. In short, the fixed stars being prodigiously removed from the earth, which is but a point in relation to this enormous distance, we can say without appreciable error that all the straight lines Kri< ra .1 I6 Lessons in Horology. drawn from the earth to the Polar star are parallel to each other, which moreover conforms to experience. Since the Polar star is found almost exactly on the prolongation of the axis of the earth, the straight line that we imagine drawn from any point on the globe to this star is parallel to the axis. We can then say that the latitude of such a place as A, which is in reality the angle A O E, is represented by the angle B A C; practically the angles A O E, and S A Care equal by having their Ndes perpendicular each to each. We see then that the latitude of J a place is equal to the height of the pole above the horizon. In J order to determine it with greater exactness it would still be neces- sary to take note of the distance of the Polar star from the axis and ' of the refraction of the luminous rays. 16. The means employed in order to determine the longitude by direct observations are scarcely practicable, therefore we use in preference a marine chronometer whose daily rate is known. Let us suppose that a ship is one day at a certain point, which we will designate by A, that we have made the observation of the , General Principles of Cosmography, 17 passage of the sun across the meridian, and that we have noted the advance or the delay of the hour marked by the chronometer at this instant. Let us call Q the number of hours, minutes and seconds shown by the watch at the exact moment of the passage of the sun across the meridian. One or several days later, we repeat this observation and we find, let us suppose, a new correction that we will designate by C. The difference between Q and C, divided by four minutes, will give us the number of degrees that the ship has advanced in one or the other direction of longitude. In short, since the earth takes twenty-four hours to accomplish a complete rotation of 360 degrees, it would require ^^ which is four minutes to cover one degree. We understand that it would still be necessary to take account in the calculation of the daily rate of the chronometer ; this value is an average based on a great number of former observations. All the accuracy of this method depends then on the absolute regularity of the daily rate of the marine timepieces. The values C^ and C are algebraic values, that is to say of quantities preceded by positive signs ( + ) or negative signs {—), since the chronometer can be behind or in advance at the time of the passage of the sun across the meridian. Let us determine now which of these signs belongs to the advance of the timepiece and which will designate its retardation. In the first place, it is evident that this choice is arbitrary ; thus, at the Observatory of Greenwich, they have adopted the negative sign for the retardation and the positive sign for the advance. This choice seems at first sufficiently normal, but at Neuchatel one is placed at a different point oi view. When the chronometer is slow at the passage of the sun, in order to find the true time we must add to the hour indicated by it the correction C; this value should then be preceded by the positive sign, while when the chronometer is fast at the passage of the sun we must subtract the correction C in order to obtain the correct time ; in this case then, this value should receive the negative sign. It is this manner oi viewing the matter which has led, at the Observatory of Neuchatel, to the adoption of the sign -f fc slow, and — for the advance of the chronometer \X. As we have said that a chronometer never follows exactly mean time, its daily rate therefore should be determined in an 1 8 Lessons in Horology. observatory : because this daily rate must be reckoned with in the determination of the longitudes. Let us designate by A this mean, value, and since it can be fast or slow let us precede it by the negative or positive sign. The chronometer la observed at the moment of the passage of the sun across the meridian ; this obser- vation giving the true lime it will be necessary to deduct the mean time from it Let us call B the diSerence between the mean time and the true time (equation of lime) and let us determine the sign of this last value. Since it is desired to bring back the correction C to the mean time we will argue that, if the mean time at noon is fast, the true time is slower than the mean time and the value B should receive the sign +, on the other hand, if the mean time is slower than the true time, B would receive the sign — . Let suppose now that a ship leaves a seaport whose longitude is E^ degrees west of Paris. The day of departure we have ob- served the passage of the sun and obtained a correction Cq . The correction C'^ of the chronometer on the mean time will be for this day C\ = C„ + B^ at the end of A^days we repeat the observation of the passage trf- the sun, and we will obtain a correction C between the time of th^. chronometer and the true time : the correction C between the' time of the chronometer and the mean time will be expressed by C' = C+ B. The difierence D betn-een the time of departure and the time of the place where the vessel now is will be Z> = ( r+ J) — (C„ + J„) - NA. Reducing this value to minutes, we will have the longitude B in degrees by the division r 18. Let us take a numerical example. The longitude of Havre west of the meridian of Paris being;! a" 13' 45". let us imagine a vessel leaving this port November 2, 1 1893. The time shown this day by a marine chronometer at the moment of the passage of the sun across the meridian is 11 h. General Principles of Mechanics. 4J m. 42 s. The equation of time for this date is + 16 m, 21 s. We will then have noted the correction C'^ — 11 h. 43 rn. 42 s. + 16 m. 21 s. = 12 h. o m. 3 s. Four days after, a new observation shows that at the moment of the passage of the sun the chronometer indicates iih. 28m. 57s. For the 6th of Novem- ber, the equation of time being + i6m. 15s, the new correction will be C= II h. 28m. 57s. + 16m. 15s. = II h. 45m. 12s. The diSerence \ ZJ- i3h. o m. 3 s. — iih. 45m. \i%. — NA I subtracting D=i\\a. 51 s. - NA. Supposing that the mean daily rate of the chronometer be A 0.5 s, we will have We will have then the longitude sought by y performing the division we will obtain £ = 3° 4a' 13" + 3° ij' 45" = 5° 56'. The ship will then be at noon on the sixth of November gitude 5° 56' west of Paris. II. General Prindplei of Mechuiici. Forces. Any cause which produces or modifies the move- ment of a body is a. force. A force can hs power or resistance, that is to say, it can, without losing its active character, act in the same manner as or contrary to the movement. Such are the effects pro- duced by animated beings, by wind, steam, waterfalls, etc. Passive forces exist naturally and can partially or totally destroy motion, but are incapable of producing it; such are, among others, the effects produced by friction, the resistance of the air, etc. 20. We can estimate very accurately the greatness of forces by their eflects. The value of a force can always be represented by a weight, as the kilogramme or gramme, which would make equilibrium with it. Thus the force exerted by a man in order to ^Hlt a car in motion can have its greatness measured by a certain 20 Lessons in Horology. number of kilogrammes. Let us suppose, for instance, a cord fastened to the car and passing over a fixed pulley placed before the vehicle ; it we suspend weights to the free end of the cord and increase them until the car commences to move, the total oi the , weight will give us the measure of the eflort put forth by the maaiS in order to produce the movement desired. I 21. As a general rule, we give the name of violive force to any ' power which puts a body in motion, and, on the other hand, that of resistant force to every active or passive force in opposition to this movement. 22. Without being able to define the nature of forces, thfr sensations which they invariably produce in us give us immediately an idea of their intensity and of their direction. The directions of forces are represented by the straight lines along which they tend to move the body to which they are applied. It is suitable to represent their intensity by lengths which are pro- portionate, the result is that we can submit forces to the ! mathematical processes as any other quantities. 23. The point of application of a force is that part of a body on which it acts direcdy in order to change the state of motion or, of rest of this body. 24. The line of direction of a force is that along which it tend* to make its point of application advance. 25. A force capable of replacing by itself alone a system cj forces acting on a body is called the resultant of all these forc**s' These last are called the components of the only force able t replace them. 26. The trajectory is a line which the movable point follows^ The movement is called rectilinear or curvilinear, according as tl trajectory is a straight line or a curve. 27. Law of inertia. Experience has established a law 1 which all bodies are subject and which constitutes a fundamental principle of mechanics. This law known under the name of "prin^ cipie of inertia'"'' can be defined as follows : A material body cannot put itself in motion if it is at rest and,' reciprocally, if it is in motion it cannot of itself modify its movement, 28. Definition of mechanics. Mechanics is the science of forces and their effects. Its object is to find the relations of the forces which affect a body or a system of bodies causing this body or this system to take a certain movement in space. Reciprocally beingj General Principles of Meckania. ai given a body or a system of bodies, to find the motion that this body or system of bodies will take in space under the action of given forces. This general problem comprehends the one in which the forces make no change in the state of the body or of the system, a particular case in which we say that the forces are in equilibrium. Thence comes the division of mechanics into sialics or the science of equilibrium and dynamics or tlie science of motion. We can still study the movements of bodies, considering only their direc- tion, intensity and duration, in leaving out the matter of which the bodies are formed and the forces which produce or modify these movements. This study forms a part of mechanics to which is given the name of kinevialics, which can also be called geometric mechanics. 29. notion. Motion is uniform, when equal distances are traversed in equal times. We call the space traversed in the unit of time velocity, we will have then, designating velocity by v, space by s and time by /, whence we have \ = \ and t = - 30. The motion is called variable if the spaces traversed in any equal times are unequal ; that is to say, when the speed of the ixidy is not constant during the entire duration of the motion. 31. When a moving body traverses certain equal distances In equal times, without fulfilling the same conditions for parts of these distances, we say that the motion is periodically uniform. Such are, for example, the motion of the earth around the sun and the vibratory motion of a pendulum in small amplitudes. 32. The motion is uniformly variable when the velocity of the moving body varies equal quantities in equal times. The acceleration is then the quantity which the velocity varies ig the unit of time. If in uniformly variable motion, the velocity increases the acceleration is positive and we say that the motion is uniformly accelerated. If the velocity diminishes, the acceleration is then negative and the motion is said to be uniformly retarded. 33. The motion of a body which falls by the action of its weight is uniformly accelerated. In this case we designate the leration due to the weight by the letter g; this value is constant p... ^■rI< a Lessons in Horology. for the same place and in our regions g = 9.8088 m. This v^uj represents twice the distance traversed during the first second bjfl a body falling freely and without initial velocity, 34. Rotary motion. A solid is animated with a movement of rotation on an axis when each of its points describes a circum- ference whose plane is perpendicular to the axis and whose center is found on this axis. In this movement any two points of the body, describe, in the same time, similar arcs, that is to say, of the sanM number of degrees ; but the lengths of these arcs are diflerenl and should be proportionate to their distances from the axis. Let e and e' be the arcs traversed in the same time by two points » and m' (Fig. 4) situated at the distances r and r' from the axis O rotation, we would have The movement of rotation is uniform if, in equal times, a poiq of the body describes always equal arcs. The velocity of such a raotitw can only be determined by con^ sidering at the same time th6 path traversed in i second by i point of the body and the distance of this point from the axis of rota- tion. In order to avoid this double data, we consider the points whidt are at the unit of distance from the axis and we call the angular velocity the length of the arc scribed in i second by a point ng. 4 situated at the unit of distan< from the axis. Let w be this arc, we will have, for the velocity v of another point situated at the distance r from whence we conclude w ^= ^- Consequently, the angulai velocity is obtained by dividing the space which is traversed \ any point in i second, by the radius of the circumference which it describes. 35. Hass of a body. The quotient of the weight of a fjody in any place on the globe, by the acceleration due to gravity at this General Principles of Mechanics. 23 place is constant. This value is what is called in mechanics the mass of the body. This quantity, which is of a particular nature since it is nothing else than a quotient, can be subjected to calcula- Q just as any other quantity. In designating it by M, we have ^^H M ^ — from whence Mg = iV and g — ^ ^^H 36. The product Mv of the mass Moi a moving body by the ^^■kxuty which it possesses, takes the name of quantify of motion. H ^ ^^piict of the intensity of this force by the path traversed by its point of application. In other words : The work produced by a force constant in magnitude and in direction is represented by the product of the intensity of this force by the projection of the distance traversed by the point of applica- tion on the direction of the force. *Thus, the path and the force being in the same direction, we have (Fig. 5) Work of a Force. 37. Deflnition. We call in mechanics work of a force the pro- ^ FY. A B. If the path A B and the force F have not the same direction, we will project in this case the path A B q\ ewill have (Fig. 6) W^ FY, A C the direction A F and Let us observe that the projection of the path on the force is IS the angle B A C becomes smaller ; the work will then I greatest when A B and A F will have the same direction, ! angle B A F becomes greater, the projection diminishes, and and ^ 38. Work of a force tangent to a wheeL On imagining the movement of rotation of the wheel to be very small, we can admit that the movement takes place along the tangent Let us call F the force and s the space traversed in its direction, we wDl have for this slight displacement W = FY^s. When the wheel will have made a complete revolution, the path s will have become a circumference and we will have the work for one revolution expressed by W^ Fy. j^r. 39. Unit Of work. We have chosen lor uriil of work that which is the product of the unit of force by the unit of distance, that is to say, in meclianics, of the kilogramme by the meter and in horology, of the gramme by the millimeter. We have given to this unit the name of kilogrammeier for the machines and of grammilli meter for the more delicate pieces of horology. As abbreviation, we will designate by the letters gr. m. this last unit which we will make use of in the entire extent of this course. If, for example, the force is 3.5 grammes and the distance 0.4 milli- meters, the work of the force will be B^= 3.5 X 0.4 = M gT. m. 40. Active power. A weight P which falls from a height h generates a certain work that we can represent by the product We find in mechanics that every body falling from a height k is animated by a velocity v which is connected with the height k by the relation i/> = 3 gh, from which expression we can draw '•- f,- Replacing in the equation of work Ph, h by this last value, we will but — being the mass w of the body, we will have at length Ph^\fnvK General Principles of Mechanics, 25 The expression i m v^ has received the name of active power. We can then say that the active power of a body in motion is half the product of its mass by the square of its velocity ; or, also, that the mechanical work which imparts a certain velocity to a body is equal to the active power which animates that body. We give the name of active force to twice the active power ; we have then Active power = \ tnv'^. Active force tn v^. 4L Every body in motion is capable of doing work. In effect the body has a mass m ; it has a velocity v, since it is in motion, con- sequentiy the product ^ m v^ gives us the value of the work Pky to which the velocity v corresponds ; we can therefore say that every body in motion is capable of producing work. Moment of a Force. 42. Let us now imagine two cylinders of different diameters turning around an axis O (Fig. 7) and let us admit, for example, that the first is three times as great as the second. Let us suspend weights at the ends of light cords wrapped around each cylinder, in such a manner that each of these weights acts in a contrary sense to the other. In order that equilibrium may exist in this system, we will find that the weight fixed to the small cyl- inder should be three times as great as that which is fixed to the large one. In this manner, if we turn the cylinder one revolution, one of the weights will rise ^ while the other will fall ; the weight p tra- il P LiP versing a path represented by 2 r X 3 its ^^' "^ work will be The weight P traverses a path 2 tt X i, producing at the same time a work Since we have 3 / = P we can admit (a) 2 TT X 3 / = 2 'T X I Z', and there will be equilibrium, because the mechanical work of one of the weights is equal to that of the other. The equality of the 26 Lessons in Horology. work of these two weights will exist also when we make the cylin* ders describe only a fraction af a revolution : as small as this fraction may be. Dividing the equation (a) by 2 tt we obtain 3 / = I P. The figures 3 and i are the respective lengths of the radii of each cylinder ; this radius takes the name of lever arm and the abovtf product of the intensity of the force by the lever arm is called thd moment 0/ ike force. Summing up, we have just examined the state of a body which can turn around a fixed point ; to this body are applied two forces whose work mutually counteracting, produces equilibrium. We call such a system a lever. In every lever, lor equilibrium to exist it is therefore necessafjF for the moments of the two forces in action to be equal. Imagining the system in motion, under the action of an exterio* impulse, we will find the work of the forces on multiplying theif moment by the angle traversed ; in this new condition of the body the work of the forces will then be equal. 43. Lever. Pracdcally, a lever is a solid body, movaUfi around a fixed point and acted upon by two forces tending t<I AV.„ .-.B*- n*. 8 make it turn in contrary directions. Fig. 8 represents a '. in which O is the fixed point. P and F the two forces, lever arms of the forces P and F are the distances from the fixt point O to the two forces, that is to say, the perpendicular O A and O ff dropped on the direction of these forces. General Principles of Mechanics, 27 From what we have said before, equilibrium will exist when the moment of the force P will be equal to the moment of the force Fy that is to say, when FXOA' = PXOB', which can be written FOB' P~~OA' The two forces should, therefore, be in inverse proportion to their lever arms. Fisr. 9 44. One often distinguishes two kinds of levers : In the lever of the first kind the fulcrum is situated between the points of application of the two forces ; in the other, Fig. 9, this fixed point is situated at one of the extremeties of the body. From the theoretical point of view this distinction is useless and the con- ditions of equilibrium of the lever apply to all cases. Transmission of Work in Machines 45. We give the name of machine to every system of bodies intended to transmit the work of forces. In order to explain in what manner this transmission is effected it is necessary to enter into some details. The relative movements of the different parts of a machine are not determined only in direction but also in intensity. Generally the movements are periodically uniform (31) ; the speed is put in harmony with the requirements of the industrial work to be produced without its ever attaining the limit at which the solidity of the machine would be endangered. A6* Different forces act on a machine in motion, which can be divided into three classes : 1st Motive forces. These are those which act in the direction 98 Lessons in Horology. ^H of the movement of the parts which they operate ; it is consequent^ Uj these that is due the motion of the machine. 2d. Useful resisting forces, those which the materials on which the machine operates, oppose to the movement of the parts which act on lliem ; it is these then which we desire to overcome. 3(1. Passive or hurtful resisting forces which arise from the movement of the ditlerent parts of the machine to oppose this movement. We have already seen that they are due to the friction of these parts among themselves or on foreign bodies, to shocks which can be produced between these parts on account of sudden changes in speed and to the resistance of the air. CunBidering the motive forces as positive, since they act in the: direction of the movement, the useful resistances and passive rCBiHtanccM will then be negative. If we suppose the system KUimatcd by a uniform movement, the algebraic sum of the work of ull the forces for any given time will be null, since the gain or the loM of active power is null, and we will have, in designating by Wn the work of the motive forces, W» the useful work and Wp the work of the passive forces : J/-. -W„ —W), = o, Ironi whence W,» --= HI + Wp, which shows lis that, the movement being uniform, the motive worto 1b equal to the useful work, augmented by the work of the passive IiirccB, When in any machine this formula is verified, we say then that there is "dynamic equilibrium," When the movement of a machine is periodically uniform, the gnin or the loss of active power is null only for a whole number ol [>eriod» : for this time we still have Wm -- IK. + n'p Wc say then that the machine is in " periodic dynamic equi- llbrlnm;" lhi» is the ordinary state of machines, not only on Aect^Hlll o( ihc aliapc ol their |n»rts, but because of the variations: miMT or Inw grrttl in the iuoii\-e fortes, iind espedally in the; Thtw M', U nlwnj-s inferior to »*« ; that is to say, a machine- remlont |m* iwHiil work than the motive power api^ied, because ihc tt\uk ol iho [>ttWLi\v rciiat*«ices is ncwr null General Principles of Mechanics. 29 At. Calling P the motive force acting on any machine, and Q the useiul resistance overcome by this machine, E and c being the spaces traversed by the points of application oi P and of Q in the direction of these forces and in any equal time, at the beginning and at the end oi which the speed of the machine is the same, the equation of dynamic equilibrium gives, supposing first that the pas- sive resistances are null : PE^ Qe. or^ -- | From the equality between the work and the power and that of the resistance, it follows that for the same motive work P E, according as the force ^ may be multiplied by J^, )4, 2, 3, etc., the space e will be divided respectively by the same numbers ; from whence comes the maxim well known in mechanics: "That which we gain in force we lose in speed, or what amounts to the same thing, in distance, and reciprocally." The preceding proportion enables us to calculate any one of the four quantities P, Q, E, e, when we know the three others. For any simple or complicated machine, il the question is to find the resistance Q that a power P can overcome, we determine the spaces E and e traversed in the same time by the points of application of the forces P and Q. E and e are any distance whatever if these points of application have uniform movements, but we take them corresponding to a period if the movement of the machine is periodic. When the machine is constructed, by putting it in motion of any sort, we determine the values of E ; we deduce those of e from the relations of the spaces traversed by the different parts which transmit the movement of the point of application from P to that of Q. Let us suppose that the resistance to be overcome Q be 100 kilogrammes, and that it is desired to determine what will be the power of P. neglecting the passive resistances. We commence by determining the corresponding values of E and e, as has just been shown. Let £^ 2.5 m., and ^^o.Som. ; replacing the letters by their values, in the preceding equation we will have i P= — ^j ' ^ 32 kilogrammes. It we had known P we would have been able to determine Q, as have just done tor P, Lessons in Horology. 48. In machines, especially in industrial machines, the passive resistances are so considerable ihat we cannot neglect the work that tfacy absorb ; the dynamic equilibrium is then expressed by W„ = IVu + IVp . For a certain displacement of the parts of the machine, the work Mffl IVu Wp will be valued as in the preceding case ; thus /* being the power, Q the useful resistance, R, R' the difTerent {tassive resistances, and E, e, i, i" . , , . the corresponding distances traversed in the same time by the points of application in the direction of these forces, we have PE^ Qe + Ri-^ R'i' ^.... 49. It may happen that one or several hurtful resistances come from the shocks between the parts of the machine. The work absorbed by these resistances is no longer valued by the product of a force by the distance that its point of application traverses, but by the loss ol active power due to the shock, and this loss, valued in units of work, enters into the second member of the equation as the other hurtful works R i R" i'. . . . By the aid of the preceding equation, knowing in a machine two of the three following works : the Wn ^ P E, the Wi =Q e and the W^= R i -^ R' V ^ we determine the third. 50. Ordinarily one decides to set up a machine capable of producing a given useful work. WH = Qe It is then necessary to determine the Wm =^P E, capable d producing not only this useful work but of overcoming also the secondary resisting works. One should then commence by calculating this hurtful work, which is done by determining the values of the diSerent passive resistances R, R" . . . . in function of Q, and afterward Wp in function of H^« . Having Wp and IV^ , we can determine the value of Wm expressed us has been said in kilogrammetere and in grammilli- mcters, 5L The motive work Wm being represented by loo, the useful and hunful works H^ and Wp being for example. 75 and 25, the loM is then 25 for 100; we say in this case that the product of the nmchinc is 75 per cent. If it were possible for the loss to be nothing, the product would be 100 per cent ; this fact can never be General Principles of Mechanics, 31 realized, which renders absolutely illusive the hypothesis of perpetual motion. The product of a machine rarely passes 80 per cent. ; it is nearly always much inferior to this limit. In this preliminary study we have desired to establish a basis which is nothing more than the enunciation of some fundamental principles of mechanics. In the course which is about to follow, we will make a constant use of them, and all their developments will be found in the text. CHAPTER I. General Functions of Clocki and Watches. The OiollatioDs of the Pendulum and th^ Relation to the Motive Force. 52. We know that in clocks and watches time is measured by the periodically- uniform movement of the pendulum or of the balance wheel. History relates that Galileo, while yet young, was struck with the regularity of the pendulous vibrations of a candelabra in Cathedral of Pisa. He studied the laws of these oscillations and used a pendulum later on for his astronomical observations. This instrument, in its primitive simplicity, presented two difSculti when the astronomer left his pendulum to itself, after having diverted it from the vertical position, the oscillations which ■ produced having at first a certain amplitude, diminished litde by little, then finally ceased entirely. He was then obliged, from time to time, to give an impulse to his pendulum. The second of these difBculties was the necessity for him to count the number of these oscillations. It is said that he charged a servant with the execution of these two functions. Now, the mechanism of the clock performs unaided these two functions with a regularity that the man could never achieve direcdy, 53. Let us seek, in the first place, for the causes which make the oscillations of a free pendulum constandy diminish. When a pendulum is moved from its position of equilibrium O A (Fig. lo), the attraction of the earth, which was perfectly neutrahzed by the resistance of the point of suspension O, is no longer so in the oblique position O B. It would cause the ball to descend vertically if the cord did not force it to describe the arc of a circle; at each instant of its returning course the speed of the pendulum increases a small quantity until it reaches e the vertical position O A. From there on the inertia, or, if one prefers it, the velocity acquired, forces it to continue in its motion and makes it describe the arc A ff; from that instant also gravity, acting in the contrary direction of the motion, tends to stop it The velocity diminishes constantly, and would become nul the moment when the ball would arrive at a height equal and General Functions of Clocks and Walckes. 33 symmetrical to that from whence it started, if there were no other forces than those of gravity which would act on the pendulum. These forces which exert their action in the contrary direction to the motion, are resistances of the suspension and of the air ; it is then to these that is due the diminution of the amplitude of the pendulum's oscillations. IE these forces could be suppressed, the motion would be perpetual. 54. There are two ~ manners of suspending a pendulum — by means of a knife edge and by means ol flexible springs. The knife-edge sus- pension is made in such a manner thai the friction is very slight, without its be- ing, however, completely annulled. This kind of suspension can be used in regulators whose amplitude of oscillation is generally small. For this purpose a sort of knife blade slightly rounded on its edge, and working in the interior of a hollow cylinder, is fixed transversely to the pendulum rod (Fig. ii). The knife edge, as the hollow cylinder, should be made of exceedingly -hard iterial and thoroughly pohshed. The knife edge can then be irded as a pivot of very small dimensions. (We will see later on that the work of friction is proportional to the pressure and to the greatness of the amplitude. ) The spring suspension consists in terminating the upper end of the pendulum rod by two short blades of steel securely fastened on the other end to any fixed piece. In this system, which is also very much used, there exists a loss of force resulting from the distorting of the blades. 55. Concerning Ike resistance of the air, we admit that it is in direct relation to the largest transverse section of the body and to square of the velocity with which it traverses the atmosphere ; Lessons in Horology. it depends, moreover, whether its form is more or less tapering. The work of this force should be proportional to the cube of the size of the amplitudes. 56. Now that we know the nature of the forces which act on the pendulum during its vibrating motion, we can determine their work and establish the relation which connects them with each other. The motive work IVm developed by gravity during the descending half oscillation, is equal to the weight P of the pen- dulum multiplied by the projection of the arc B A (Fig. lo) on the direction O A ol the force ; therefore, by the length Ab (37). We will write then U'n. ^ Py.Ab The resisting work, that is to say the work of the forces which act in the contrary direction to the motion, is composed of two distinct forces : 1st. Of the work of gravity developed while the pendulum traversed the half oscillation ascend- ing, therefore the weight P multiplied by the pro- jection of the arc traversed A B' on the direction A; let us represent this work by the formula F's '1 W«=Py.Aa 2d. Of the secondary resisting works arisbg from the resist- ances of the suspension and of the air. Knowing the lengths of the arcs A B and A B', we find the work of the secondary resisting forces by multiplying the weight P of the pendulum by the diflerence of the projections Ab — Aa or ab ; we will then have Wp = Py. ab The motive work should be equal to the sum of the resisting works (46) ; we will therefore have or substituting PXAb = PXAa->rPXab 57. For the oscillations of the pendulum to preserve the same amplitude, it is therefore necessary that at each of these oscillations it must receive an impulse whose work should be equal to P X a&, 58. Since the secondary resisting work increases with the amplitude of the oscillations, it is necessary that the impulse, or what we should call the work of the maintaining force, should be greatest when we wish the pendulum to traverse the largest arcs. ^^« Genera! Functions of Clocks and Watches. 35 We see also that the more we diminish the friction of the knife edge and the resistance of the air, the less maintaining force is necessary. We diminish the resistance of the air by using a pendulum ball of high specific gravity, because for such a weight the section which traverses the air is smaller. The pendulum can also be suspended under a glass from which the air has been exhausted. 59. In order to maintain the oscillations of the pendulum in clocks we use most frequently motive forces produced by a weight, a coiled spring or an electric current. The two first will be the subject of a detailed study in the iollowing chapter. The Oidllation) of the Balance and <itxai Relation to the Motive Force <0. The motion of the pendulum cannot be employed for leasuring time, except in instruments which can maintain a fixed position. In portable timepieces we utilize the vibratory motion of an annular balance, mounted on an axis and furnished with a spiral spring. 61< This spiral spring is a thin blade of metal, of sufficient length, wound on itself in the form of the spiral of Archimedes, or of a cylindrical, spherical or conical helix. In each case, one of the extremities of this blade is fastened to the balance wheel and the other to a piece fastened to some part of the watch. When we place in a watch movement the balance wheel fitted only with its spiral, there is found a position in which the elastic force of the spring exercises no influence on the balance. The latter is then in the condition of repose. When we move the balance from this position in either direc- m the elastic force of the spring tends to bring it back to the lint of repose ; there are then produced oscillations analagous to those of the pendulum. This oscillatory motion is very useful for measuring time and has the advantage of being suitable for employment in all portable timepieces. 62. Suppose A (Fig. 12) the point of repose of a balance wheel ; if the latter be moved from that position the angle A B = a., and if at the point B it be released, thus allowing the elastic force of the spring to act on it, this force will impart to it a movement of rotation whose speed will increase up to the point A. ling that point, the spiral will exert a force contrary to the 36 Lessons in Horology. direction of the motion and tending to stop it. It it were possible to produce such a movement without there being any passive s acting on the balance wheel, the latter would traverse a new angle, A O 6 = n., \ then would come back to 3, m and so on indefinitely. ' It is not so in reality, for there are a number of resisting forces which act on the balance and which pre- J vent it from arriving at b. I These forces are : ' 1. The friction of the balance pivots. 2. The resistance of the air. 3. A loss of force re^l J., J, dent in the spiral, the true cause of which is not abso- lutely defined but the existence of which can be perfectly established. These secondary resisting forces have the effect of diminishing each oscillation a small quantity, which is represented in the figure by the angle B O b. Calling a' the angle ff A, wg have B' O b ^ a. — a:. If, as we have done for the oscillations of the pendulum, w« designate by Wm the motive work exerted by the spiral while th< balance wheel traversed the angle a, Wa the resisting work pro- ceeding from the spiral during the second part of the oscillatiop ; therefore, while the balance wheel traverses the angle a', and Wp the secondary resisting work of the passive forces, we would.' obtain the equality (46) Wm = Jtu + Wp, The work of the maintaining force should be, both for the; balance and for the pendulum, equal to the secondary resisting^' work, if you wish to preserve the initial greatness of the amplitude of the oscillations ; otherwise expressed, the work of the maintaining force should be equal to the work of the force of the spiral while the balance wheel traverses the angle a — a.'. I General Functions of Clocks and Watches. 37 63. We can admit that the resisting work increases with the amplitude of the oscillations, as we have shown for the pendulum, and conclude that more motive work would be necessary to tra- verse larger arcs than for smaller ones. <4. We use exclusively for motive force in portable timepieces the elastic force developed by a spring enclosed in the interior of a ^^K»rt ■ called the barrel. Tiiis piece, generally toothed, turns lOund an axis, and this action is conveyed to the balance wheel by >ecial mechanism, which we are going to pass rapidly in review. Wheel-Work. lb PuTpoie in the MechBoiiin of Clocki aad Watche*. 65. The motive force, not acting directly either on the pendu- or on the balance wheel, is first transmitted by a system of toothed wheels or train of gearings that is called in technical lan- guage the wheel-work or the transmission. This force, thus trans- irted, is received by a mechanism which is the escaperneni ; it is 3S Lessons in Horology. this last whose function it is to restore to each oscillation of the j pendulum or balance whee' the loss of force, ff'-. — (K. . occasioned by the secondary resisting forces. 66. When a weight is used as motive force, that weight is sus- pended to the extremity of a cord unwinding from a cylinder fixed concentrically on the axis of a toothed wheel. This wheel A (Fig. 13) gears in a second wheel much smaller than the first and which is called a pinion, on which is fixed concentrically a second toothed wheel B, which in its turn gears in the pinion &, and so on to the last pinion, on whose axis the escape wheel is fastened. The same thing takes place when the motive force is that of the spring in the barrel. In this case the barrel gears directly into the fit pinion a. bX, The diSerent wheels of the wheel-work in watches the following names : 1. The barrel. 2. The center whee! * (large intermediate wheel). 3. The third wheel (small intermediate wheel). 4. The fourth wheel (seconds wheel). 5. The escape wheel (escape wheel). The pinions carrying the four last mobiles take the same name: as the whee! to which they are riveted. 68. The mechanical worlc of the motive force is then trans- mitted by the wheel-work to the escape wheel. This transmission cannot be effected, however, in a complete manner, because part of the force is absorbed by the friction of the gearings and of the pivots, by the inertia of the moving bodies and sometimes also by the defects resident in the gearings. 69. Beside the transmission of the force, the wheel-work should fulfil another function : this is to reckon the number of oscillations that the pendulum or the balance wheel executes during a deter- mined time and to indicate this number by means of hands on a properly -divided dial. We must, therefore, combine the relation of the numbers of teeth in the wheels to the numbers of leaves in II Id n» of the 1 •WealfB hon lb* KiipnIU ointuD (>? Nausbktel. Tbe uunM vt Incxs und of ■: ohmleie and ■hmild bt nnlsHid Vy lbs (Iillowtait, ithlcb Bra in bt of ihHs two mohtUw 1 " CanWf wheel " for Ihe drat and '■ int«riaf General Functions of Clocks and Watches. 39 the pinions, so as to make this indication conform to the division of time. Thus, the center wheel carrying on its axis the minute hand should complete one rotation during one hour, and the fourth wheel carrying the second hand should make one revolution each minute. (The hour hand is carried by a wheel forming part of ail accessory wheel-work, which will occupy our attention later.) Eicspementi. 70. Several kinds of escapements have been constructed, difiering more or less from each other, but whatever they may be their function consists always in restoring to the pendulum or to the balance wheel the speed which the passive resistances have made them lose. The most perfect escapement will be the one which will effect this work by altering as little as possible the dura- tion of the oscillation. Since the movement of the balance wheel as well as that of the pendulum is an oscillating movement, the escape wheel is arrested ^ during part of the oscillation ; it is only when the balance or the pendulum has traversed a determined arc that the wheel becomes free and is put in motion. During this time it acts either directly on the balance, as in the ' ' cylinder ' ' escapements or the ' ' detent, ' ' or on an intermediate piece, as in the "anchor" escapements. After having traversed the angle of impulse determined, the wheel arrested anew until another disengagement. The manner in ich this arresting is produced differs according to the kind of ipement. Tl. In most of the escapements the action of each tooth of the wheel corresponds to two oscillations of the balance wheel or pen- dulum. Thus, in a watch, the balance wheel executes 30 oscilla- during one complete revolution of a wheel of 15 teeth ; in a , the pendulum makes 60 oscillations during one revolution of escape wheel of 30 teeth. 72. To recapitulate, the study of the functions of horological ichanism can be divided into four principal parts, which are ; Power — study of motive powers. , Transmission — study of wheel-works and gearings. . Reception — study of the escapements. , Regulation —study of regulating and adjusting. ^Hfa ai ^Klic ^^ca] CHAPTER 11. Maintaining or Motive Forces. The VlafiA u a Motive Forcr. T3. We will adopt in the beginning as units in the calculations, the millimeter as unit of length, the gramme as unit of weight and of force, which gives us for the unit of mechanical work the grammillimeter. We will choose the second as the unit of time. 74. Among all the forces which are used in horology in order to maintain the oscillations of the pendulum, the weight is at once the most regular, the most simple to obtain and the one whose intensity can be regulated with the greatest facility. 75. If a certain weight P (Fig, 13) is suspended at the end ot^ a cord wrapped around a cylinder the radius of which increased by , half the thickness of the cord is equal to r, the work of this force while; the cylinder executes one revolution will be expressed by (38) Z' X 2 T r. Dividing this work by the number N of oscillations that the, pendulum executes during one revolution of the cylinder, we will have as quotient the mechanical work developed by the weight during one oscillation of 'the pendulum, thus : We know that a part of this mechanical work is lost during its J transmission to the pendulum : calling W^ this last work, we.J should have the equality W /• — Ifl ^ Wp, in which we will replace W Phy its value, thus : We see then that the determination of the work which the pendulum receives at each oscillation ( I'l^ ) depends also on the knowledge of the work lost during its transmission by the wheel- work and the escapement. We understand, consequently, the | diiliculty that there is to determine the motive work, since this j work does not depend alone on the weight and on the dimensions j of the pendulum but also on the resistances to be overcome during an oscillation. Here are. however, two calculations taken from practice which \ will aid in more firmly fixing the ideas on this subject ; r Maintaining or Motive Forces. 4! M. First Calculation. — The motive weight of a regulator beating seconds is 2000 grammes ; this weight is suspended at the end of a cord which unwinds from a cylinder, with a radius of 15 millimeters. What will be the work produced by this weight during the unit of time ? The mechanical work efiected by the weight while the cylinder b executes one revolution will be ■ EocKj X 2 IT X 15 -= "88496 gr.m. 1 A wheel.<4 is fastened to the cylinder (Fig. 13) gearing in a pinion which carries on its axis a second wheel B, which in turn gears into a pinion &, this last pinion carrying on its axis the minute hand should then execute one revolution an hour. The numbers of teeth and leaves of these moving bodies are distributed in such a manner that the pinion f> executes 45 turns while the cylinder makes one ; consequently, one revolution of the cylinder takes 45 hours or in 45 X 60 X 60 = 162000 seconds = A^ We will then obtain the work produced by the weight during one oscillation of the pendulum, by the application of the formula, ^^e » /> - -V^ - f^Z -= ^-'^3 Er.T We will show the manner of calculating the work lost during Bie transmission when we treat of the questions of frictions, of the inertia of the wheels, etc. ; for the present, let us admit these calcu- lations as made and adopt for this special case the value ItVu = 0.413 gr.m, e will then have tfm — ICb = Wp , or 1.163 — 0,413 = 0.7s gr.m. The weight of 0.75 grammes, exerting its action on a dis- ice of one millimeter, is then sufficient to keep up the oscillations 01 a pendulum whose weight is about 6500 grammes. The ampli- tude of the oscillations is 2° 6'. 7r. Although that which follows is a little outside of the problem which we have just solved, let us profit, however, by the data that we possess to calculate further the angle B O A — B' O A (Fig. 10). 'his adjunct to the preceding solution does not, moreover, lack in interest. ^^his; Following an equalion previously established (56), the work of the force | capable of maintaining the oscillations of a pendulum was expressed by IVp = PX a 6. " ^ = -6500" = 0.0001154. The length of a simple jjendulum beating seconds is about 994 milli- meters for our latitude.* Let us suppose that the entire weight of our pen- dulum is assembled at a single point, the distance from the center of gravity to the center of suspension will then be equal to the length of a simple pen- dulum beating seconds. We will have ^i = 994 — 994cos/^05 I From Fig. 10 the difference A b ~ tracting then the two foregoing equations, a 4 = 994 cos .rJ £' - whence follows since the angle A O B\i equal, in this case, to half of 2° (/, which is 1° 3', we can write, after having completed the calculation of the second member t& the equation : Cos A OB' — cos 1° 3' = 0.000000116. In order to determine the value of the angle A O B ^ A B', we can find in a table of natural trigonometrical lines the difference between the cosineSn of the angles 1° a' and 1° 3'. This difference is 0.0000053 ! ws will theO' have the proportion, then AOB — AOB'= i.-f. 78. Second Calculatimt. — A clock from the Black Forest, such as those that were manufactured in iai^e quantities during the years between 1S40 and 1850, runs under the action of a weight 625 grammes. This weight descends in 24 hours from a height of 1350 millimeters. What is the work produced by this force during one second ? The work produced during the descent of the weight will be W= 635 X IJ50 in 34 hours ; during one second it will be 24 X 60 X 60 = times less ; therefore * LiUiuilc ct Neuoliilil Maintaining or Motive Forces. 43 We see that this clock requires a much greater mechanical work than that of the regulator of the preceding example. This difference becomes still more obvious if we compare the two pendulums. The weight of the pendulum of the last clock is only 8 grammes, while the pendulum of the regulator weighs 6500 grammes. Although we could not, at this time, compare two clocks, whose pendulums have neither the same length, nor the same weight, nor the same amplitude of oscillation, we note, however, that the regulator requires much less motive force than a small ^ clock of the Black Forest. The Barrel Spring u n Motive Force. 79. These springs are thin blades of properly- tempered steel ; I ^H^they are of a sufficient length and coiled up in spiral form i ^■'interior of the barrel. One of their extremities is fastened to the ^H- wall of the drxim and the other to the hib, which is a cylindrical ^^Vpiece adjusted on the arbor of the barrel or forming part of it, ^^"When one holds firmly either the barrel arbor or the barrel, and ^^ causes tlie one of these two pieces left free to turn, the spring begins to wind around the hub and manifests a certain force from its extremities, which tends to bring it back to its first form. When the arbor is made fast, the force displayed by the spring has then the eSect of causing the barrel to revolve. 80. The place occupied by the spring in the interior of the barrel should be equal to half the disposable space. 81. Measurement of the Force of a Sprlnf. The force developed by the spring is susceptible of measurement. For this purpose let us adjust on the barrel arbor a graduated lever arm, along which a certain determined weight can slide. While holding the barrel in the hand, let us set up the spring to the point that we wish to study, one turn for example ; let us endeavor then to produce equilibrium by sliding the weight along the lever arm. When the two actions, that of the weight on one side and of the spring on the Other, neutralize each other, equilibrium is produced, and it is then evident that the efTort displayed by the spring is equal to the eflect produced by the weight. This last effect will be perfectly determined when we know the size of the weight and the length of the lever arm, at the extremity of which it exerts its action. We know that in mechanics the moment of a force (4a) is the —product of the intensity of this force by its lever arm. 44 Lessons in Horology. The moment of the iorce of the weight will give us then the moment of the force of the spring. 82. If the lever of the preceding experiment has not its center of gravity on the axis, it will still be necessary to take account of the effect produced by the weight of this lever, which cannot, practically, be reduced to a simple geometric line. In order to determine this we must find the distance of the center of gravity of the lever from the axis, and multiply this value by the weight of the lever. We then add this product to the moment of the force previously obtained. Let us suppose, for example, that a weight of 20 grammes, suspended at the extremity of a lever arm 200 millimeters long, makes equilibrium with the elastic Iorce of a barrel spring. The product 20 X 200 = 4000 represents the moment of the force exerted by the weight. If, moreover, the weight of the lever is 7 grammes, and the distance from its center of gravity to the center of the arbor 143 millimeters, the moment of the force e.xerted by the lever will be 7 X 143 = io°i- Adding this value to the moment of the force of the weight, we obtain the moment of the force of the spring that we will designate by F, then ^=400 This - 1000 = 500Q grammes the approximate value of the moment in round numbers. of the force of the spring in a watch of 43 millimeters (ig hnes). Let us remark that generally these levers are furnished with counter weights combined in such a manner that the center of gravity is found on the axis. 83. The number 5000 that we have just obtained, signifies that the spring considered is capable of making equilibrium with a weight of 5000 grammes suspended at the extremity of a lever arm equal to the unit of length, therefore i millimeter (Fig. 14). 84. Examining in this manner the force of a spring, we will prove that it varies very much according to the number of turns that it is set up. Experience proves in fact that the moment of the force of a spring being, for example, at its maximum point of tension, 5000 grammes, this moment constantly diminishes, and will not be more than about 3400 grammes when the barrel will have executed four rotations around its axis. Mainiaining or Motive Forces. 45 85. We understand then thai the imperfections of the primitive watches being known, the ancient horologists should have sought means for correcting the inequality of the action of the motive spring, and that for this purpose they should have invented the ingenious arrangement of the fusee, which will be explained later on. This corrective is really almost entirely abandoned, and is seldom used except in marine chronometers ; in pocket watches it has become useless in ]>rDportion as the improvements in the con- struction of escapements have come into use, and as the isochronism of the oscillations of the balance wheel has been obtained. ITheoretica] Study of the Moment of a Sprin£'j Force. 86. Coefficient of Elasticity. When a body receives an exterior effort, the molecules which compose it tend to follow the direction of this force ; they approach each other or separate themselves, the one from the other. The result is a force equal and opposite, iFhich tends to make the displaced molecules recover their former sitions. This property, common to all bodies in different degrees, is illed their elasticity. I LessoTis in Horology. According to the effort exerted, the molecules approach or leave each other ; the first case is an effect of compression or contraction, the second is an effect of tension. ST. The reaction is always equal to the action ; we can then measure the elastic force of bodies by the exterior effort which is applied to them. The following experiment wilt explain this assertion : 88. Let us secure one of the extremities of any vertical rod, to the other extremity we suspend a weight (Fig, 15). This rod from that moment undergoes a certain elonga- BHSMl'MI l^'on, and we can prove that the molecular efEort devel- 1^^^^ oped is equal to the weight producing the elongation. The elongation of this rod will depend on the size of the force P, on the length of the piece in its natural state, on the cross section of this piece being assumed the same throughout, and finally on the material of which it is composed. By experimenting on a rubber band we can see that under the action of a force P, the transverse section of the body diminishes at the same time that the elonga- tion is produced. This regular diminution on almost the entire length of the band does not take 'place uniformly near the two points of fastening. Therefore, it is necessary to take the elements for the experiment sufficiently removed from these points in order to eliminate a source of error which would influence the final result. Let us take again, for example, a rod of iron, whose transverse section is i square mm. ; we have measured the distance between two marked points sufficiently removed from the points of fastening ; let L be this length. We suspend from the lower end of the rod a weight P, and we measure anew the length between the two marks ; we obtain then an elonga- fig. IS tion /. Experiments made in this manner have de- monstrated that, provided the load P does not surpass a certain limit, / remains proportional to the load. Supposing now that the experiment were physically possible, let us determine what should be the load P that could produce an elongation equal to the original length L. We call thia Maintaining or Motive Forces. particular value of P the coefficient of elasticUy of the body ; we will designate it by the letter E. The elongation being proportional to the load, we have Thus, in the case of an iron rod, whose original length L was looo millimeters, if we suspended from it a load P of looo grammes, : will find an elongation of 0.05 mm., which gives as the coefficient of the elasticity of iron, 89. The elongation / is inversely proportional to the transverse section of the body ; thus for a section of surface s the formula I above will become „ . 90. When the coefficient of elasticity is known, it is easy to determine the value of the force exerted by the molecules of a body subjected to the action ol an exterior force by the relation The jfrad The p — --'-' 'he fraction j represents the elongation per unit of length ; this .ction should remain very small for this formula to be exact. The quantities £, s, L are constant ; P and / vary together. The same formula expresses the relation which connects a force P of compression to the contraction /. which results from the action of this force, when the piece compressed does not bend. We will give then to P and to I the signs + and — , -|- for the forces of tension and the elongations, — for the forces of com- pression and the contractions ; the formula then becomes general. 91. Variation of the Coefficient ol Elasticity. All watchmakers know that after having forged a piece of brass, the elastic force of the metal is increased. In hammering this body one diminishes its volume, but one cannot change its weight ; the molecules which com- pose the piece are forced together, and the specific weight of the metal will be increased. This simple fact shows us that the coefficient of elasticity of solid bodies should vary with their specific weight. When a watch (not compensated), regulated to a certain temperature, is exposed to a higher temperature, it loses about 48 Lessons in Horology. 10 seconds in 24 hours for each degree centigrade. The spring is expanded by the effect of the increase of temperature, its mole-- cules are separated from each other ; the specific weight of the metal has diminished at the same time as its coefficient of elasticity.. The reverse takes place when the watch is observed at a lower temperature than that to which it had been regulated. It does not appear that the coefficient of elasticity of sted undergoes a great variation by the effect of tempering and that of reheating. A piece of steel in fact changes its dimensions ^ little by tempering. It has been proven that by tempering va- water a piece of steel stretches about , j^j of its original length, but that this elongation is lost when the piece is reheated to the blue color, the specific weight of the steel not being modified, the coefficient of elasticity retains the same value as that which it. possessed before tempering. 92. We give here a table of the coefficients of elasticity of some bodies employed in horology. The figures below are taken from the " Almanac of the Bureau of Longitudes."* VkIuw of the Coeffidenti of Elutidty, E. (Hard Bronze : 90 Coppei 20599 7358 7589 {Ordinary Phosphorous ozjo Laveissiere 9061 Copper 12jM9 Berry Iron 2097a Rraw ■ / 31 Zinc 9277 «"^' 168 Copper 9395 f 18 Zinc 1 German Silver : -j 61 Copj jper :kel I Nicf Gold 8132 Palladium '1759 Platinum 17044 ,rid»dP,.,i„™: {»■/«"»„} 15518 21426 8735 ed In kiloKrinimeB 1 inhoroloKyisgen ■Itclty In 1 menfly eap fore laSKlOO Maintaining or Motive Forees. 49 93. Limit of Elasticity. If we submit any rod to the action of difierent loads, we note that as long as the load does not exceed a certain limit, proportionate to the transverse secdon of the body, the rod resumes its original length, after the removal of the weight. By increasing the weight so as to pass this limit, the elongation only partially disappears or perhaps does not disappear at all. This limit is called the "limit of perfect elasticity" of the body con- sidered. If we continue to increase the weight, the elongation becomes more and more apparent and at length a "rupture" is produced. The limit of perfect elasticity is very slight for certain metals, such as lead, red copper, aluminum, etc. Iron, even, does not possess a very great limit ; on the other hand, steel, when it is tempered, increases its limit of elasticity by suitable reheating. This reheating, known under the name of "spring temper," cor- responds to the bright blue color. It would he of great use in horology to know the exact value of this limit of perfect elasticity of hardened and tempered steel ; these experiments have not yet been thoroughly studied and the data is consequently lacking. For the present we will confine our- selves to the results with which the practice of horology furnishes us. 94. Homent of tbe Elastic Force of a Spring: Subjected to a FiexiOQ. Let A^ B^ be a spring of circular form, of rectangular Lessons in Horology. I I 50 section, of thickness e and height k. Let us imagine blade of spring be divided in the direction of its length into a certain number of fibres, one of which, especially, situated in the middle of the body, is called the "neutral fibre" for the reason that it does not change its length when the spring undergoes a flexion. When this blade is bent in such a way that the radius of the neutral fibre diminishes (Fig. 16), the fibres interior to this undergo a shortening, while the exterior fibres are lengthened. Let it V be the distance of any fibre from the neutral fibre. + 11 if the fibre is on the exterior and — vM it is on the interior rf the neutral fibre. If r„ represents the radius of the neutral fibre the unchanged position and e the angle that the two radii ending' at the extremities A^ and B^, form between them, we will have the length /-o of the neutral fibre by the equation io = r^e and the length Z,', of any fibre whatever whose radius is 7 will be L', = (r„ + z') 6. If now, one of the extremities of the blade is fastened and we bend the other, making it traverse an angle ± «, the radius of the neutral fibre will diminish if a. is positive, that is to say, if it adds to the angle e. If, on the contrary, the extremity Bq is bent in the. opposite direction, the angle fl becomes smaller and we have ii this case a negative : the radius of the neutral fibre will increase. The length L^ of this fibre has not changed by the flexion we will have then in this new position, r being the radius changed position of the fibre, Lo =r (B + a), we then have r. e = r (B + d), from whence The fibre taken whose length is L\ has become L', = (,r+!>) (fl + . Replacing r by the value above, we will have ^'. = (|R:\ +•)<' + "• and in working out Maintaining or Motive Forces. 51 the elongation / of the fibre considered will be obtained by taking the difference between lengths L' and L\ then i' — i'o = / = r„ e 4- 1- e + ;- a — fo s — K e and simplifying (,) i = v 0.. This elongation is positive ; but it will become negative for v negative ; that is to say, for the interior fibres there will be a shortening. There will also be a shortening if v is positive and a negative. If these two values are negative, their product is posi- tive and we have an elongation. Let us remark that the elonga- tion / is independent of the radius r^ of the neutral fibre and that consequently the spring can be of any form. 95. Let us now determine the moment of the force exerted by two opposite fibres, situated at equal distances + v and — v from the neutral fibre and let us suppose that the flexion of the blade may have been effected preserving the center O to the changed position of the spring ; that is to say, that the blade may have taken the position A B (Fig. 16). The exterior fibre, which has been lengthened by the flexion, will tend to return to its first length and will act with a force P whose value is represented by (90) ^'-^^ which we can replace / by its equivalent v a. (94), which gives us le interior fibre tends to lengthen out and will exert this same ce in the opposite direction, therefore — ^- The moment M P o\ the simultaneous effort produced by the the action of these two fibres will be equal to the sum of the pro- ducts of each of the two forces by their respective lever arms r \ v ■ V. Therefore M P ^ P {r-^v) — P {r~v). (3) MP^2 Pv. This value is then' independent of the radius of curvature of 'tL spring, that is to say of the distance from the exterior attaching int of the blade to the center of the barrel. Lessons in Horology. Replacing in the equation (3), P, by its value (a) v Let us now regard the section s, of the fibres considered. The cross section ol the spring being imagined rectangular, of height A and thickness e we will have, supposing first that the blade i is divided into a definite number of fibres, 10 for example, the section of one of these fibres J = 0.1 f X A, since the thickness of one fibre will be in this case the tenth part of the total thickness of the spring. Let us admit, what is not absolutely exact, that each separate fibre acts through its middle part, that is to say, that the distance V from the middle of the fibre nearest to the neutral fibre be 0.05 C' (Fig. 17), the distance of the middle of the second o. 15 e, that of the third 0.25 e, for the fourth 0.35 e and then for the fifth o.i Since in the equation (4) the term v is to the second power we should raise each of the five preceding values to the square and i find the sum of them. We will then have = 0.0025 '* = 0.0225 '' = 0.0625 *' = 0.1225 ^* ^ o, 2025 e* = 0.4125 e* 1st fibre w =: 0.05 e . v-^ = 0.05= 2d " 1/ =^ O.ISf , . . v^ = 0.15^ 3d " V = 0.25 e . . ■ w" = 0.25* 4th " V = 0.35 e . . ■ ■o'^ = 0-35' 5th '■ V = 0.45 e . . . v^ = 0.45 » The sum of I MahUaining or Motive Forces. 53 Replacing now the values determined of s and of i'* in the equation (4) we will have t- ,, „ £ a 0.1 « A. 0.4125 f' Sura M P ^ 1 -. — - — This formula represents the moment of the force of all the ten fibres considered, therefore of the endre spring, while the formula (4) gave the value of the moment of two fibres only, the one interior and the other exterior, to the neutral fibre. Designating by F the preceding sum, we will have, by performing the operations indicated „ c . 1 (5) F= °-">^'-'- . We have obtained the coefficient 0.0825 by dividing the blade of the spring into 10 fibres ; ii we had supposed it divided into a very large number of fibres, we would have arrived at a value very nearly approaching 0.08333, say, ^. We would have then this case. (6) '' = — ;fi-'^ 96. We have arrived at this last form, which is the exact 'one, only by approximation. Integral calculus furnishes us a means of effecting the above calculation with an absolute exactitude and ior an infinite number of fibres. (*) take up again the equation (4) MP = ignating the inliDitely small thickness of a fibre by dv the section s s ^. dt/.k; e~J "■'" = ! — Utlne the irtntlo tl Replacing f by J r, that Lessons in Horology. isy, taking the integral between the limits^ -^ = A 2 "■ 97. In the preceding formula, a is the angle which we have made the free extremity of the spring describe from the position where the elastic effort is null, to the point which we wished to study. Thus, when we have turned the tree extremity of the spring one revoiudon, the original number of revolutions will be increased by r, etc. We can then estimate the angle a by counting the number of turns which the spring is wound up at the moment considered, not forgetting to deduct from tfiis figure the number of turns which the spring makes if placed unconfined on a table. Let n be tbiS'— diSerence, we will have H we can write the formula (6J H 98. On calculating the moment of the force of a barrel spring \ by means of this equation, and on then comparing the result obtained with that which the experiment gives (S3), we generally find a slight difference. This difference proceeds essentially from the following causes : ist. As we have already stated, the value of the coefficient of elasticity of the spring with which we are engaged could be perceptibly different from that which we have admitted in the calculation. ad. It is difficult to measure exactly the thickness of the spring s a slight error will give a considerable difierence in the result Thus* for a spring of o. i3 mm. an error of -j^ °^ ^ millimeter wl influence the result one-sixth. 3d. The transverse section of the blade is rarely a perfeel rectangle ; the spring is often concave on the outside and convex aU the inside. 4th. The calculation supposes the spring to be perfecdy freCj but complicated effects are produced when it is shut up in the barrel. When it is wound around the hub of the arbor there is only a I certain length of the blade which is freed from the coils pressed against the drum of the barrel. The moment of the force should therefore be calculated according to the length of the blade freed. J Maintainmg or Motive Forces. 55 5th. When the spring is wound up to a certain point, the coils of which it is composed deviate from the circular form and spread out to one side ; there is thus produced a decomposition of force, one of the components of which is directed towards the center of the barrel and is transformed into friction. We can add a similar defect which is produced at the center and which on combining with the exterior fault can diminish, or, in certain cases, increase the moment of the force of the spring. 6th. Considerable friction is produced between the coils of the spring ; the oil which we are obliged to use to reduce friction produces a slight effect by its adhesive force. 99. Example for the Numerical Calculation of the Formula (7). The dimensions of the spring for a watch 43 mm. diameter (19 lines) being the following : Thickness, ^, =0.18 mm. Height, hy =- 3.6 mm. Length, Z, = 650 mm. to calculate the moment of the force of this spring. When the elastic effort of this spring is nothing, that is to say, when it is placed perfectly free on a table, it makes 5 turns. Coiled in the interior of the barrel and pressing against the drum, it makes 14. The development of this spring being 6 turns in the barrel, a half turn is given for safety, and we will have, according to what has been said, « = 14 + 5.5 — 5 = 14.5 turns, when the watch is completely wound up. Let us take the coefficient of elasticity, E = 23000000. The formula (7) can be written thus : -, E h e^ le n 6L replacing the letters by their values, we have „_ 23000000 X 3-6 X o.i8^ X 3-1416 X i4'5 6 X 650 Effecting the above calculations we find that F = 5640 gr. for « = 14.5 turns. Lessom in Horology. The simplest way of effecting the above calculation is by means of logarithims. We give below the method of such an operation : Log. E ^= Ic^. 230000CX) = 7.3617278 log. e» = log. 0.18* log. h = log. 3.6 log. , ^ log. 3.1416 = Log. numerator Log. L — log. 650 = 2 + log. 6 = - 0.7658175 - 3 = 0.5563025 ^ 0.4971499 6. 1809977- J 8129134 m 7781513 ■ Log. denominator 3 Log. numerator 6 — log. denominator 3 5910647. ■ 1809977 5910647 + log. K = log. 145 = I 5899330 = log. 38S.985 161 5680 3 7513010 = log. 5640. The preceding calculation shows that the moment of the force ol the spring is 388.985 gr. for a number w ^ i. For « ^= 14.5 ii is 5640 gr. When, on account of the running of the watch, the barrel has made one turn, /( will have diminished one unit and will only have 13.5 as value ; the moment of the force of the spring will have diminished 388.985 gr., or, in round numbers, 389 gr. We can then form the following table ; /" f or « ^ 14.5 = 5640, the spring is entirely wound up, /^, '■n^i3.5^525i| the barrel has made one turn. J-'j " K r= 12.5 = 4862, the barrel has made two turns. F'f " n =^ II. 5 ^ 4473, the barrel has made three turns. F't " n = 10.5 ;^ 4084, the barrel has made four turns. 100. Inequality of the Elastic Force of the Spring:. The moment F of the force of a spring is then greater when the watch U completely wound than when it is about to stop. This fact has been already demonstrated to us by experiments (84). It b necessary to confine this inequality of the motive force witilin the narrowest limits possible. Let us note for this purpose tin the numerical calcularion of the formula {7) we have succes- ^ replaced n by h — r, « — 2, w — 3, and a — 4 ; in this last I IflW Ibe traldi is at the instant when it is about to stop, if the barrel I with stop works. But the ratio between n and n — 4 1 is smaller, it will be proper, in order to diminish t {nqnafityof the force, to give to the number of turns, a, of 4 Maintaining or Motive Forces. 57 the spring, as great a value as possible. The following demonstra- tion will better explain this idea. lOL Since we can use springs producing the same initial moment offeree ^0' ^"^ whose dimensions and number of turns, «, are different, we understand that F^ may vary in certain cases much less than in others. Let us suppose, for example, two springs producing, when wound up, the same moment of force F^ = 4000 ; the first having a number of turns n = 10, the second a number » = 20. For the firat we would have in the formula (7) the value and for the second this same valut f When the two barrels will have executed one revolution, the num- ber of turns of the first will be n, = 9 and that of second »i = 19. will then have successively : First Case. F^ ^ doo X 10 ^ 4000 F^ ~ 400 X 9 — 3600 F^ = 400 X 8 = 3200 Ff = 400 X 7 ^ aSoo Fi ^= 400 X 6 = 2400 Second Case. F^ = 200 X 20 ^= 4000 Fi = 200 X 19 = 3800 Fj = 200 X 1 8 — 3600 F^ = 200 X 17 = 3400 /^, — 20a X 16 = 3200 It is thus easy to see that the moment of the force diminishes in the first case much more rapidly than in the second, and that, the force of the second spring approaches much more a constant value than that of the first. It is best, then, to give to the number of turns, a, the greatest value possible. For a given spring this number cannot, however, exceed a cer- tain value determined by the limit of perfect elasticity of the steel, which cannot be exceeded without setting or breaking the spring. This limit depends on the elongation per unit of length ~ of the exterior fibre. We have had (94) h can be written 58 Lessons in Horology. In the preceding numerical example we have had the following values : v^\e^ 0.09, a = 3 « fi = 14.5 X 2 ■». Z = 650 mm. We will then have J_ ^ 0.09 X 14-5 X 2 ■• L 650 and on performing the calculations X = °''"^"' """■ We can admit this value of -j- as the limit allowing sufficient security, and established by practice. 102. It must not be forgotten, however, that the nature of the I steel, the manner in which the springs are manufactured, hardened | and tempered, can materially modify this limit. The springs are hardened and tempered in circular form, witll | about 100 mm. radius ; they are then worked in a spring tool and by this operation the fibres undergo quite an unequal elongation, since, from the first circular form, they pass to a spiral form whose radii of curvature for the interior coils are much smaller than thosM , for the exterior coiis. It follows, from this operation, that the exterior fibres : elongated more in the inner coils than in the outer ones. It is4 for this reason that the springs break more often interiorly than 1 exteriorly. 103. The form which a spring has on leaving the hands of the maker is very variable and it can be with difficulty represented 1^ a general equation. ' This primitive form is not preserved after the watch has run a certain time : the spring "gives " a httle at first and finally ends by , taking a form which it keeps permanently. This last form is the one | ft-hich should be taken as the starting point from which to determine the angle inn giving the degree of flexion in the formula (7). Starting from this position, we can admit that the elongation per I unit of weight -3- is much the same for the whole length of the spring, j 104. Length of the Spring. Since we cannot unfold a spring, , in order to measure its length, without modifying its interior strue- j ture, it is convenient to have at our disposal a simple formul enabling us to calculate the value L. Supposing the spring coiled in the interior of the barrel ; will admit that the radius extending to the interior coils may t Maintaining or Motive Farces. S9 equal to ^ ^ in the position of the spring at rest, R being the interior radius oi the barrel. We can, without great error, substi- tute circumferences for coils and obtain the length of the spring by multiplying the mean radius, by 2 w N', on deciding to designate by N' the number of coils in the spring when it is pressed against the side of the barrel. To this value must be added the length of the end of the spring which is detached ■om the coils so as to be hooked to the hub, which is about ; would have, therefore, the length L ^ -i -^ R (I N' + \). We have, for example, in the calculation of the length of a iring for a watch of 43 mm. the following values : R = 8.8 mm. and N' = 13 coils, [tepiacing the values, we will have Z = 2 X 3-1416 X 8.8 (J X 13 + i); I whence Development of a Spring. I05. When a watch spring is put in the barrel it is wound on itself and forms a certain number of coils, the outside one of which presses against the side of the barrel and the succeeding ones against each other, taking the form of a spiral of Archimedes, with grada- tion equal to the thickness of the spring. The inner end of the blade is disengaged abruptly from the coils and is fastened to the hub of the arbor. In order to "wind" the spring, we can hold the barrel and turn the arbor several turns until the spring may be entirely wound around the hub, with the exception of its outer end, which remains fastened to the side of the barrel. It is evident that the number A'' of revolutions which the arbor has been able to make is equal to the difference between the num- ber of coils that the spring has in these two extreme positions. 6o Lessons in Horology. Let N', be the number of coils which the spring makes wt is pressed against the side of the barrel ; N" the number of coils when wound around the hub. We will have then N^= JV" — A". In order to simplify the calculations, let us neglect the inner and outer ends of the blade which are disengaged from the coils, and admit that the space occupied by the spring in the two positions a cylindrical volume. Let us designate, moreover by /?, the interior radius of the barrel ; r, the radius of the hub ; r", the radius extending to the interior coil of the spring when it is pressed against the side of the barrel (Fig. i8) r", the radius extending to the outer coil of the spring when is wound around the hub (Fig. 19) ; e, the thickness of the spring. We can write ' = and i consequently (i) A'= N" — N' = We can find the value of r" ^H and CO ^^B We can find the value of r" in functions of R, r* and r ^^v observing that the surfaces occupied by the spring in the two poai- ^H tions are equivalent. ^H When it is pressed against the side of the barrel, this surface b ^L 5 = » (^'--^^'), Maintaining or Motive Forces. when it is wound around the hub 5 = ir (.*» -»•')■ "herefore, r*' — r' = J? ' — r", 1 whence ^ V R^ hibsdtuting this value of r" in the equation (i), we have (J) N = ~ (^ R^ ~ ^^ ^ r^ - r - R + r'^. 106. Let us note that when \ tquation „ : obtain the value of € in the we find that it differs from the real value, which is always less. This difference arises from the fact that on account of certain inequahties of the spring, the coils of the blade do not strictly superpose, 107. Maxiraum of iV in Terms of r'. The equation (2) indicates that for a barrel whose interior radius R, the radius of the hub r and the thickness of the spring e, are determined, the num- ber of turns of development N varies with the radius f' extending to the first inner coil of the spring in its state of rest. This last radius, r', depends on the length of the blade. Let us apply, in the first place, this formula to a numerical mple and use the following data ; Interior radius of the fjarrel . . ^ ^ 3 Radius of the hub r ^ i Fraction — ^=13, The equation (2) will become, after replacing the known uitities by their values, N^ I 9+1 - r"" The smallest value that the radius ■/ can have is r' = r = \ lod its greatest value may be r' ^R = i. In the first case the spring has a number of coils sufficient to com- etely fill the space between the side of the barrel and the hub, and n the second this number of coils is nothing. The reasoning shoiv^ 62 Lessons in Horology. that in these two extreme cases the development of the spring' woa be nothing, which the apphcation of the formula (2) also proves. Replacing successively in this same formula r' by 1,1, i.; 1,3, etc., up to 2.9, we can form the following table : y N '■ N r' N 1-7 4.76 a-4 5.96 I.I 0.84 1.8 5.3 2.5 5-67 i.t 1.63 I.9 S-56 2.6 S' 1.3 ».37 2 5.84 2.7 4-5 1-4 3.0S j.r 6.03 2.8 3.5 1-5 369 a. a 6.13 2.9 3.09 1.6 4.6 2-3 6.11 3 ° One sees that the maximum number of coils in the develop- ment ol the spring is given by a radius r' equal to about 2.2. In practice, | of ^ has been adopted as the value of r' for the reason that the regularity of the power from beginning to end increases with the length of the spring. 108. The chIcuIus enables one to determine the exact value of the maximum of N in function of r'. Let us take up again the equation (i) : N=\ (^' R^ - r" -i- r' - r - /i + r'), N and r'. Let us differentiate this equation J?' - r" + r' = ^, in which the two variables by placing d N= ^ {ki* dz + d r'). Replacing .; and d a hy their values, it becomes f JV= i ( • \ 1 ")■ Maintaining or Motive Forces. qnating thb derivative to zero : : VR* - Ijtaising to the square we have (3) .'=V^4^. Substituting this value of r' in the equation We see that the tnaximum oS N o s when one has — v^ Since it is the custom, in practice, to make the radius of the hub equal to one-third of the interior radius of the barrel, we can place Jf = 3 and r = i and we will have /I P 109. In order to represent graphically, the equation (2), let us rder it to two rectangular coordinate axes (Fig. 20) and lay oS on &e axis of the abscissa the values of r ', and on the axis of the ordinate the corresponding values ol JV, and connect the points thus obtained by a curved line. One sees that on making the unoccupied space of the barrel equal to the part occupied by the spring one does not obtain the Vmaximum turns of development of the spring. 110, If one divides the interior of the barrel, giving ^ ^ to the part occupied by the spring, ^ R for the empty space, i R for the radius of the hub. N= \ C/F^T^ I - 3 + ») Lessons in Horology. and A'= HV6-a) == 7(^-44 -a) = 7X0.44 Consequently, if 7 = 13 one will have N=^ 5.7 turns r = H ' N=b.\ '• V = 15 " " " J^= 6.6 " UL Diameter of the Hub. The custom of making the radius 61. ! :;! 5! . of the hub equal to one-third of the interior radius of the barrel has. been established by long practice. When one leaves the hub greater than this value, one does not obtain a sufficient number ol turns of development of the spring, and when one makes it smaller the spring is apt to break, or if it is too much reduced in temper it • It H be« not to Bilend tbe figures of the aaann root of 6, becnuse od laooont oribt Interior put of the ipHug IhIdk ilUchcd to (he hub, one thereTore Iokb man rmdllr ■ litU* of the develDprneot. r Maintaining or Motive Forces. 65 Cases present themselves, however, where, in a given barrel, one desires to introduce a thinner spring than that which one has been in the habit of using and it may then be asked if one could not reduce the diameter of the hub, in order to obtain a greater development of the spring. Let us then examine this question. It is known that when one submits a rod of steel to a tension, the transverse section being equal to one square millimeter, this rod lengthens, and that when the load which produces this elonga- tion reaches a certain value, a rupture is produced. Now, expe- riments made with spring wire have demonstrated that the wire breaks if the weight reaches a value of 135 kilogrammes. Let us note that the wire which was used for this experiment not hardened. Let us first seek again the elongation / per unit of length which the rod underwent at the instant immediately preceding the rupture. We have (88) El W {"aking E= 23000000, P ^ 135000 gr, , L ^ i, we will place / ^ '350O0 ^ 13s 33000000 23000 rhetice I = 0,0058695 mm. On the other hand, let us calculate the elongation of the exterior fibre of a barrel spring as it is admitted in practice and compare the two results. For this purpose let us take the spring of a watch of Tim. (19 lines) which has hjrnished proof of being able to bear desired flexion. We will have the elongation per unit of length from the ula (94) !he thickness of this spring is o. 18 mm., consequently 'In the interior of the barrel and pressed against the drum this spring had 13 turns and 5 outside of the barrel ; moreover, after beiDg wound, it was set up 5. 5 turns. With this information we find ■ a = (13 — S + 55) X 2 IT = 37 IT. 66 Lessons in Horology. The length of the spring is 600 mm, ; one will have consequentlf Comparing the above figures, we can establish the astonishing result that the exterior fibres of a spring can sustain an elongation per unit of length twice as great as that which produces a rupture by tension. This fact cannot be explained by the supposition of a superior quality of steel to that of the metal composing the rod which broke under the action of a weight of 135 kilog. ; because this last steel was certainly of the first quality. It must then be admitted that the exterior fibre of a spring does not break, when the elongation that it acquires is equal to that which produces rapture by tension, for the reason that it is retained by the interior fibres. In the presence of this fact one can admit as the limit of elongation that the exterior fibre of a spring can bear without breaking, the value _ One remains within this limit in making the diameter of the hub equal to one-third of the interior diameter of the barrel and in using a spring making 13 turns in one-third of the interior radius of this same barrel. When one desires to use a thicker or thinner spring, one must in this case determine the diameter of the hub with relation to the thickness chosen. Thus the interior diameter of the barrel which we have used in the preceding experiment being 17.4 mm., the diameter of the hub was then iM - 58 mm. Dividing this diameter by the thickness of the spring 0.18 one arrives at the conclusion that the diameter of the hub should be 32 dmes the thickness of the spring, in round numbers. If the diameter of the hub is smaller than this proportion, the spring runs the risk of breaking or, if it is too soft, of setting. Work Produced b^ a Spring. 112. One determines the mechanical work which a spring pro- duces at each oscillation of the balance wheel, by dividing the work Maintaining or Motive Forces. 67 displayed by the spring during one turn of the barrel, by the num- ber of oscillations made by the balance wheel during this time. Let F = 4800 be the moment of the force of the spring of a watch whose balance executes 18000 oscillations an hour ; we will obtain the mechanical work effected by the spring during one turn of the barrel by the product H' = 4800 X a w. If the barrel of 80 teeth gears into the center pinion of 10 leaves, the number of corresponding osciilattona will be the mechanical work during one oscillation will then be k 4800 X Jw '*'"= ?5x.8ooo -«-»°WEr.mm. If, on the other hand, the watch only beats i&zoo oscillations hour, one would have in this case Wm = X 16J00 One sees then that the work of the force received by the balance wheel at each oscillation is increased by diminishing the number of these oscillations. Let us suppose again that the balance wheel I executes 18000 oscillations, but that the pinion of the center wheel ) has 12 leaves in place of 10, one would then have 4 S00 y air ' f5 X 18000 ' = 0.351328 gr.ti! One, consequently, also increases the force by diminishing the f duration of a revolution of the barrel. The Fusee 113. We know that the law of the variations in the action of a spring which unwinds, is complex, and that the force developed has widely separated limits, for a spring of the same thickness its whole length. In order to remedy this defect, the ingenious arrangement of ^e fusee was conceived long ago, consisting of a solid body whose 68 Lessons in Horology. secdonal revolution is very nearly parabolic, and whose surface" grooved with a helicoidal curve. This piece is mounted on the axis of a toothed wheel A (Fig. 21) gearing with the pinion of the center wheeL The teeth of the barrel are, in this case, suppressed, and its arbor remains constandy fixed. This arrangement permits of the complete equalization of the motive action of the spring. In efTect, when one has just wound the watch, the spring is completely coiled ; a steel chain, one end of which is hooked to the fusee and the other to the barrel, is at this moment almost completely wrapped around the spiral lines of the axis of the fusee. On unwinding itself, the spring turns the barrel, which communicates its movement to the fusee by the intermediary of the chain. This unwraps itself, little by little, from the fusee, and wraps around the barrel until there remain no more turns on the fusee. It is evident that if the tension of the spring continually diminishes, this action works in the contrary sense, always at a greater distance from the axis of the fusee. The motive work, product of the tension by the distance traversed, gives, designating the tension by P, the distance to the axis by r and the speed c& rotation by w, Prw. This work should lje constant if the speed of the wheel fixed on the axis of the fusee is constant, that is to say, if there is a uniform angular speed, and the fusee is grooved in such a manner' that the product P r remains constant. The variations of P must then be the reverse of those of r. An exception, however, must be made to the preceding, if one takes into account the friction of the pivots in the plates ; this friction, in fact, diminishes constantly as Maintaining or Motive Forces. I tbe pressure diminishes. We will, however, neglect this lactor, in ■ Brder not to complicate the following theory. I 114. To determine practically whether or not the variation of ■ the force o£ a spring is exactly counterbalanced by the form of the fusee, one uses a lever and a weight, as we have seen before (8i) ; one fixes the lever on the square of the arbor of the fusee ; the form of this piece will be exact if the weight carried by the lever makes quilibrium with the force of the spring at the same distance from the axis tor each point of the successive rotations of the fusee. 115. Calculation of the Variable Radius of tbe Fusee's Helix. Let it, be the interior radius of the barrel, half the thickness of the chain being included therein : r, the variable radius of the fusee ; r^, its initial radius (in rand r,, is also included half the thickness of the chain) ; I I, the maximum angle which the spring is wound, starting from the position where the elastic effect is null, and corresponding to the instant when the chain acts at the extremity of the radius r^ ; ^ the angle which the barrel has turned, starting from the posi- tion t ; i^ the angle which the fusee has turned, starting from the instant when the chain acts at the extremity of the radius r^. e moment of the force of the spring can be expressed by ^ Ee' h -«). A-g^' -- M fc will have F = M {i — B.). The force F' acting at the exterior of the barrel should be i the moment F^ with relation to the axis of the fusee is The values of r and of a should vary in such a manner that, in order that F^ may be constant, we should have a equal to zero for '■equal to r^ ; we will then have 76 Lessons in Horology. ^ whence R R (1) «' = e — o' 1 When the chain wraps up an infinitely small quantity, j? <^ «, 1 on the barrel, it unwraps the same length, r rf p, from the fusee. 1 One has then yP rf a = r rf p, J but, because of the equation (i), 1 On integrating it becomes 9 — n) fl-o. -/-)■ These integrals should be taken a ^ a', one will have between the limits a = o and 1 To rf a = e a and fa ^ a = 5 .' consequently, Drawing from this equation the value of a. one will have first - i «■ 4- B a -^'^ changing the signs, adding $■ to ea it becomes »• - a e a + e' = and consequently. ch member and multiplying by 2, Replacing or now in equation (i) a by this valb e, we will obtain 1 .-W" a/- -^1^ Maintaining or Motive Forces. Placing fl ^ 2 « M and p :^ 2 « m' we will have, finally, ^ 116. Namerlcal Calculation of tbc Preceding Equation. Let ^=8 mm., ''0^5 mm., 8^12X2* and let us calculate first the value of the radius r for an angle p ^ 2 ^ ; we will write, on replacing values, Log : 86 ■- - log : 96 -- W-^^-- V-- 'iveH : 1-9344985 1.9822712 ax I" — 2 Log; 5 = 0.6989700 _log:^|^ 0.976.136 1.9522273 0.9761136 Log : r' = 0.7228564 and r' — s.aSa;^^. Successive calculations will give us in an analogous manner the rilowing results, which we group in a table : For p = 2 „, r' = 5. 2827. " P = A *, r" = 5- 6195- " P = 6 -. r'" = 6.0302. " P = 8 ,, r'" = 6,5491. " p — 10 w, r'" ' = 7-2231. 117. Other Calculations. It more often happens, in practice, that one b given the greatest radius of the fusee, and that the ques- tion is to determine the variable radius, starting from this value. This problem, the inverse of the preceding, is solved in an analogous manner. Preserving the same notations as in the pre- ceding case, let r^ in this case be the greatest radius of the fusee and a the angle which the spring is set up at the instant when the a acts at the extremity of the radius r^ of the fusee. We have e moment of the force of the spring : F^ = M e, I in 'he initial position, when the barrel has turned an angle a, F=. M(t + a). ^2 Lessons in Horology, The force F* acting at the exterior of the barrel will be for the two cases : M M and the moment of these forces with relation to the axis of the fusee will be : F'\ = ^^ and /?-'' = ^ (0 + •). Making these two values equal, one has Mr, ^^ Mr or from whence one extracts As in the preceding case, we place r aT p = /? aT a, from whence ar p = — ar a. r Replacing r by its value (i), one obtains fl^ P = -4 ( e + a) ar a. and on integrating, from whence p - -^- ( e « + i a»). Transformations analogous to the preceding case will give us successively : i a' + e a -- ^ p, •« + J e a + 0« = ^ p + e», « = - 6 ± ^ 2 ?:^ p + e«, Maintaining or Motive Forces, 73 The value of a, extracted from the equation (i), is equal to a = ii - 0. r consequently, and r from whence '•• 6 = ± e ^^l- + X. r = \ R% ^ + x; or still further, by substituting p = 2 » «' and = 2 v n, (2) r = ^0 + I. 118. Numerical Calcalatton of the Preceding: EqnatioiL As an example of the application of the preceding calculation, let us determine the dimensions of a marine chronometer's fusee and let the following be the data : Exterior radius of the barrel including half the thickness of the chain, R =^ 21.7 mm. Maximum radius of the fusee, . . . . r^ = 18.3 mm. Development of the spring, « := 3.4 turns. Let us admit, that when the spring is set up one turn, the chain acts on an angle p == o, in this case then n' = o. When the fusee has made one turn, we then will have »' = i, and on replacing the letters by their values in the formula (2), we will have 18.3 4- 2 X 18.3 XI., 21.7 X 3.4 The calculation gives L(^ : (2 X 18.3) = log : 36.6 = 1.5634811 —log : (27.7X 3.4) = 1.8679386 0.6955425 — I Corresponding number = 0.4961 Log : 21.7 = 1.3364597. + log: 3-4 = 0.5314789. 1.8679386. Lessons in Horology. i-og: V 0.4961 + I log: 1.496 Log; 1.496 0.1749606 0.0874803 Log: 18.3 = Number = i.*6i45ii 0.0874803 1. 1749708 14.961 w< will then have the radius of the fusee for a nu turns ri = I ; . = 14.961 mm. Replacing, successively, in the preceding formula n' by 2, 3, 4, etc, one will arrive at the following results : ^ 7, 653 ^ 119. Uniformity of tbe SprmE:*s Force in Fnsee Watcbes. In order to obtain perfect uniformity of the spring's force in fusee watches, it is not sufficient alone to construct the fusee in a manner conformed to the data ol the preceding calculations. There are Other factors which must be taken into account, and about which we will give some explanations. In order to verify the uniformity of the force of the spring with rdation to the fusee, or, to speak in shop parlance, in order to equalize the fusee, one places the fusee and the barrel between the two plates of the watch, puts the chain in place, sets up the spring, and fastens on the arbor of the fusee the lever that we have DKOtioned (114)- Holding the movement of the watch in the hand, one then tnnis the lever a quarter of a turn and establishes equilibrium by means of weights; then one turns the lever i, 2, 3, etc., turns, taking care to notice if at each revolution the equilibrium is maintained. But practice teaches that if, in this operation, one finds an increase of force, one approaches perfect equality by further setting tip the mainspring ; if, on the contrary, tbe instrument shows a Maintaining c '"^ease of force, one lets Ike spring down. When one ass found '''s uniformity of tfie force, tlie initial position of the spring's 'snsion is preserved by marking a point on tfie pivot of the "^el arbor, and by repeating this point on the plate of the watch 9pposite to the position that the former occupies. Let us remark that it is necessary to renew this operation each le that one replaces the spring experimented with, by a new spring. 120. Let us seek now for an explanation of the effect whicli is (Produced in the preceding experiments. Let us admit that the •"adii of the spiral lines of the fusee may have been calculated for a Spring which, being wound, is set up lo turns and has unwound 3 turns at the moment when the chain is completely unrolled from the fusee. Being completely wound, the moment of the force of the Spring is then proportional to lo. AVhen the barrel has made i turn, this moment is proportional to 9. "When the barrel has made 2 turns, this moment is proportional to 8, "When the barrel has made 3 turns, this moment is proportional to 7. From the lop to the bottom, the moment of the force has diminished ^5 ^= 0.3, and in order that equilibrium may be produced, the radii of the spiral lines of the fusee must have increased in the same proportion. If, on the other hand, the spring was set up only 9 turns, the chain being wound on the fusee, we would have in this case : Spring completely stretched, moment of the force proportional to 9. The barrel has made i turn, moment of the force proportional to S. The barrel has made 2 turns, moment of the force proportional to 7. The barrel has made 3 turns, moment of the force proportional to 6, The moment of the force has then diminished | ^; J = 0.333. This decrease is superior to that of the first case, the same fusee will not produce equilibrium and one sees thus that it will be necessary, in order to have equilibrium, to set up the spring another turn. 121. We possess in this way a means of regulating, practically, the moment of the force of the spring with relation to the axis of the fusee, taking into account certain factors which have not been introduced into the preceding calculations ; the principal among them being friction. It is evident that on setting the spring up further, we increase its energy ; it is, moreover, only in rare cases that any incon- venience will result from this increase of force. 76 Leaons in Horology. 122. In establishing the theory of the barrel spring, we have admitted these springs to be of the same thickness from one end to the other, that their coiled blades remain always concentric, that is to say. retain a spiral form during their development ; and, finally, that they are always free. Practice shows that these conditions are not fulfilled by the spring inclosed in a barrel and that they cannot be so except for a free spring, such as the hairspring. Let us examine rapidly, however, these three facts, commenc- ing with the last expressed. 123. Complete liberty of the blades of the spring does not exist practically. Let us suppose, in effect, that one has turned the barrel arbof a small angle, a quarter turn, for example. In this position all the coils of the spring have not yet entered into play ; there are found a certain number which remain pressed against each other, forming part of the barrel. One should then take the acting length of the spring L, equal to the length of the spring which has become free, and the number n equal to the number of turns that this free part contains at the instant considered, diminishing this value by the number of turns that this same length possesses when it is placed freely on the table (97). Thus (Fig. 22), if a is the end of the spring hooked to the hub. and b the point at which the free coils become sepa- rated from those which remain pressed against the drum of the barrel, the length L in the equation {97) L is then only equal to the *■' length of the part a d tA the spring. Furthermore, the number n will be equal to the num- ber of coils which this part of the blade contains, diminished by the number of coils which this same length contains in the free posi- tion of the spring (Fig. 23). For example, we have (Fig. 22) Maintaining or Motive Forces. TJ fl J = af coils and (Fig. 23) a ^ = 2^ coils ; consequently. Let us note that in most watches in which the barrel makes three turns in 24 hours, one can readily admit that the whole blade ^^ becomes free, while the spring is developed these three turns, and one can employ, without great error, the above formula, without modifying anything therein. 124. Some have tried to use springs of varying thickness, that I, those whose blades increase or diminish in thickness from one end D the other. I-et us suppose in the first place that the thickness may con- stantly increase ; at the interior, the blade, being thinner, will bend more easily ; during the winding of the watch, the coils which detach themselves from those which remain pressed together, will ^^Bperform this movement in a more gradual manner than if th^- J9 Lessons in Horology. ^^^^^H thickness were the same along the whole length. On continuing to wind the spring some of the coils will be wrapped around the hub and form part of it. Thus, the moment of the force of the spring could only be determined in this case by taking the length of its iree part alone, and the value of n should also be determined accord- ing to this length. Since the thickness, moreover, is variable, it becomes difficult to determine by calculation the force of such a spring in a sufficiendy exact manner. Such springs are used to advantage in fusee watches, because they have a more concentric development and consequendy produce less friction between the coils. If, on account of the diminution of thickness at the interior, there results a greater difference between the moments of force of the beginning and the end, this difference could easily be corrected by the fusee. 125. Springs thicker at the interior than at the exterior are hardly to be recommended, for the interior part of the blade bending only with difficulty is hard to wind around the hub ; it has, moreover, the efTect of detaching a greater length of blade from the part that remains pressed against the barrel, which produces considerable friction between the coils. These springs have, moreover, a great tendency to break. 126. The principal defect of the development of the spring in the interior of the barrel is that which arises from the eccentric coiling or uncoiling of the blades ; these push themselves to one side, both at the interior and at the exterior of the spring. This is, also, an analogous fact to that which shows itself in a flat spiral without return curve. When the interior fault comes into contact with the exterior fault, the spring makes a sudden jump, producing a noise well known to watchmakers. The exterior fault can be remedied by fixing to the spring a flexible check of sufficient length, about a half turn ; this check should be made thin in the part which is fastened to the barrel, in order to permit it to follow freely the coiling up of the spring. A better remedy would be to have the last exterior half turn thicker and make it gradually thinner to suit the condidons. Stop-Work. 127. We designate by this name a mechanism fastened to 1 barrel and whose object is to stop the winding before the spring t completely coiled around the hub. ' This same mechanism alsoJ p.. Maintaining or Motive Forces. 79 Stops the running of the watch before the spring is completely pressed against the inner wall of the barrel ; its eSect then is to utilize only a part of the development of the spring, that during vhich the force is most equal. Thus the total development of the tring being, for example, six turns, ii the unwinding is arrested by the stop-work after four turns and a half, the spring will still be stretched one turn and a half when the watch stops. In most watches, in fact, this mechanism allows the barrel to make four revolutions Fig. 31 on its arbor, and if it makes one turn in eight hours, as is often the case, the watch should run for 32 hours. The most modem stop-work is what is called the ' ' maltese cross." It is composed of two pieces, the finger and the wheel.* The latter being shaped like a maltese cross, gives it this name. The wheel is placed on the barrel, where it can turn freely, while the finger is placed on the arbor. The head of this piece gears in the notches of the cross, the rounded out teeth of which can successively slip around the drcuniference of the finger. On winding the watch, one turns the barrel arbor ; the finger participates then in thiq movement and pushes, at each turn, ~" "' a tooth of the wheel until the moment when the shoulder of the filler comes into contact with the full tooth of the wheel ; the movement is then stopped, and the watch is wound (Fig. 24). During the running of the watch, the finger is stationary and the wheel, turning with the barrel, at each turn presents one of its openings in front of the end of the finger, which forces it to make a fraction of a revolution on its axis (Fig. 25). After flie four revolutions of the barrel, the other shoulder of the finger comes again in contact with the full tooth of the wheel, and the watch is stopped (Fig. 26). 128. Geometrical Construction of the Maltese Cross Stop-Work. In order to construct, graphically, this stop-work, we will suppose *Th«se pieccB lie commoatj knavn ai the male sad fenule — Xs.KS'o^ttH. 8o Lessons % that the distance O O' between the centers o( the barrel and cS the maltese cross wheel 13 known {Fig. 27). We divide this distance into five equal parts, and Irom the center O, with a radius equal to three of these divisions, one describes a circumference, which 13 repeated a second time from the point O' as center. From thiS' last center we further describe a new circle passing through the center of the barrel, and we divide this last circumference into five I equal parts. In order to get the end of the finger in one of the openings of the maltese cross, the division is commenced at the. point 1'; for other cases we wilt commence to divide at the point 0\ or any other intermediate point. The circumference of the finger is described with a radius equal to half of the distance O'; from the points i', 2', 3' and 5' will trace with the same radius the arcs A C, A' C, etc., the arms of the cross. The intersections C A', etc., will, consequendy, determine the size of the openings, the straight sides of which are drawn parallel to each other and at equal distances from the center 0'. MaifUaining or Motive Forces. 8i The essential conditions to be fulfilled in making the end of the finger are the solidity of the piece and the free action of the mechanism. We prefer to represent it by means of two arcs of circles : one m n^ whose length equals a semi-circumference, and the other n ky whose center is found almost on the point of the shoulder of the finger. Practically, the end of the finger k I should be slightly smaller than the corresponding opening of the other piece, that is to say, there should exist a certain play, to assure the free action of the mechanism ; this play will be easily obtained by taking off the sharp comers k and / of the finger. It is also to be recommended, in practice, to make the full tooth of the maltese cross with a radius O' D longer than the radius O' C ol the cut-out teeth, in order to cause the stoppage a little before the line of centers. It is, moreover, necessary to slightly round ofi the corners A^ Q A\ C, etc., of the teeth of the wheel. CHAPTER IIL IWhott^lMforiL Pvpists 4 dock ha\-« a double dittr to fulfill : firsts to transmit the move- ment drtsmg from the motive pover^ from die first mobile down to the escapement : seconds to reckon die nomber of osdDations accomplbhed by the buLmce whedL in a given time, incficatii^ this time by metins ol hjiod^ on a spaced diaL Since oa the one hand die movement o£ die fagJance wfaed is a rap^i one and on the other die motive fiocce shoold only be expended :$Iow{y. and. moreover, die wheelb carrying^ die hands ^should make certiin munbers ol tumsw accorcfin^ to given rda> tioii& one xmdersticcdb that die wheet-^vork slioaki be ai i ai^ed in such a manner jb$ to mutti^y. pro^:cessnrd^» die speed of die first mobile. Ttbb &$ why we make Ose whedb gieor edCo p^^yrms, and die aumber^ of teeth ot diese xffl&tenc msL?bSes slnaki be exsMrtly detitsrmined. Lee u:< fartS ttfr cesxfiirk tiu£ an dtns coosUaabbr ikMne a Awr the ^!eed. w^ srnnm&<& st tsue :saaK pco^ioctioii t&e uce tcansoHned to irsc ^ae resaDoit wicci <&kxa5£ ess^c ^^ecweest thie amnfcer at aoQi&ie :ry nse axcvtn^ S>5fi!? cr a ^^sarin^ ami tiar ^geoL SJ3cw:n^ nse :tHincer ct 5sea: ^^ it a wa«eL ami <8 3t rui cmic:tt ^ iat wd»:o: ± <^sa:!Si. we iw^ ^aua: ij; iar :5i^ X imiicer %. Zjs: xs <arco5e a w^sft^ .^ ^ .3C ^seci ^lescm^: 3t a rnxran m of r^ Jta^^*^ snc^ «ic3: 3:^:t3: ^ ^^ wOtteC -irwss »i*^ jstt ^ die ijsarr sssTL ^ :ait ««?* wnl ^ur'^ j^Mm^?t£ is iwp:? Whtcl- Wart. (.) - 33 turns. The preceding equation can be presented under the form If one wished to know the number of rotations completed by the pinion, while the wheel makes any number of them, m ^ 4, for instance, the number n' will become w turns greater and one could place : KferSi umii ptrair tfO w Fig. (a) toiiles I The numbers of turns made by the two ; aversely proportional to their '.bers of teeth. 131. Let us now consider train of gearings formed of fO wheels and two pinions Fig. 28), the wheel with A teeth gearing in the pinion of a leaves, on the axis of which is fastened a wheel with B teeth gearing in a pinion with b leaves. While the wheel A com- pleted n rotations, B made The number of turns that the pinion a makes while the Iheei A makes a number n, is expressed by the formula (i) in the other hand, the number of turns that the pinion b makes hile the wheel B makes n' , should be t since in this last formula one can write, after replacing ra' by its value, I If, for example, the third wheel 10 li center wheel has 80 teeth, che pinion the third wheel 75 teeth and the 84 Lessons in Horology. fourth pinion 10 leaves, one will find the number of turns that this last pinion should make while the center wheel makes i , by replacing the letters of the formula (3) by their numerical values, then The fourth pinion, then, makes 60 rotations while the center wheel makes one. Since the axis of the center wheel carries the minute hand and the axis of the fourth wheel carries the second hand, the movement will be executed, properly, according to the accepted division of time. If it were necessary to calculate the number of rotations of the fourth pinion, while the center wheel made 24, one would place in an analogous manner n" = 24 X I -^- " = 1440 tirns. 132. One could determine, in like manner, for any number ol wheels and pinions, the relation of the numbers of turns of the last pinion to those of the first wheel. This relation is always equal to the quotient obtained by dividing the product of the numbers of teeth in the wheels by the product of the number of leaves in the pinions. One can then establish, in a general manner A B CD E (5) I L Suppose, for example, we wish to determine the number of revolutions accomplished by an escape pinion while the barrel makes 4, knowing that this barrel has 96 teeth, the center wheel also 96, the third wheel 90, the fourth wheel 80, and that all the pinions have 12 leaves, except that of the escape, which has 8. We would write the formula (5) under the form : ,,. ,,,, AB CD from whence, by replacing values, //// -, 96 X 96 X 90 X 80 133. Calculation of the Number of Osdllatioiis of the Balance^ It ia generally customary to indicate the number of oscillations which the balance wheel of a watch makes during one hour, that is, while renter wheel, which carries the minute hand on its axis, makes N IVAeeZ-JVork. 85 134. We have already called attention to the fact that in most of the escapements the action of each tooth of the wheel corres- ponds to two oscillations of the balance (71). Knowing then the number of rotations which the escape wheel makes during one hour, one will easily calculate the number of oscillations which the balance executes during this same lime, by multiplying the number of turns of the escape wheel by twice the number of its teeth, an operation which can be represented, designating the number of teeth of this last wheel by £ and the number of oscillations by N, by the formula (7) N^ 2 En"'. If an escape wheel with 15 teeth makes, for example, 600 turns while the center wheel makes 1, we will obtain the number of oscil- ladons made by the balance by ;V = J X IS X 600 ^ 18000 oscillations. 135. If we designate by B, the number of teeth of the center wheel, " " " " third " D, •• " " " " " fourth " 6, " " " leaves " " third pinion, d. fourth escape E should have, according to the formula (5) ,„ ._ _ BCD . Replacing n"' by this last value in the equaUon (7), we will tain the general formula a. formula which enables us to calculate the number of oscilladons made by a balance wheel during one hour, knowing the numbers of teeth of the different mobiles. Suppose, for a numerical example, we desire to calculate the number of oscillations of a balance, knowing that 5 = 64 C = 60 D = ba £^15 Lessons in Horology. = iSooo oscillations. The application of the formula (8) will give jtf- 64X60X60X^X15 8X8X6 136. Calculations of the Numbers of Teeth. Suppose now we wish to calculate the numbers of teeth in the wheels and of leaver in the pinions, the numbers of turns or of oscillations being known. This question, the reverse of the preceding one, can have severaL. solutions ; in short, if one takes the equation J t^" _ AB CD I ' n abed ' ■ in which the relation ~ alone may be known, and in which the unknown quantities may be A, B, C, D and a, b, c, d, one sees immediately that an unlimited number of values could satisfy this relation ; the equation is, in fact, indeterminate and affords as many unknown quantities as there are wheels and pinions. In order to determine them, one chooses arbitrarily the value of some of these unknown quantities, and, in order that the result will contain no fractions, one chooses for numbers of leaves in the pinions those employed in practical use. These numbers are, generally, 6, 7 and 8 for the escape piniona,. 8, 10 and iz for the pinions of the third and fourth wheels, 10, ii_ and 14 for the pinions of the center wheels. The values a, d, c, d becoming, thus, known quantities, could be transposed to the first member of the equation, which will be. written under the fonn In order to solve this equation, it will suffice, then, to resolve all the known numbers of the first member into their prime factors and to form these factors into as many groups as there are unknown quantities to be determined. Let us take a numerical example, and suppose that the relation in the above equation be -^ = 4800. We will then have ^ g rn let us choose the following numbers of leaves for the pinions a ^ 10, i = 10, c ^ 10 and rf = 7, we will place IVAeeZ-lVork. Resolving 4800 with its prime factors, we obtain 4800 = 2« X 3 X 5' 10 = 2 X 5 and 7 is already a prime number. One will have then the total product : 3' X 3 X 5' X 7 = 4800 X 'o X 10 X 10 X 7 = with the factors of which we 1 groups : 1 form the following ' X 5 = 80 or ' X 5 = 80 " X 5' - 75 " X 5 X 7 = 70 " etc., etc. ^ = 2^ X 5* £=2» X 3 X 7 C=2*X5 D=2 X 5^ By employing other numbers of leaves for the pinions, one can niiiltiply the solutions to infinity. As proof, we could have 4800 = 3 X 80 X 75 X 7° o X 10 X 10 X 7 ^ X Bo X 5° 3X7' 137. When the relation ^ is fractional, one factors the nume- rator and denominator separately, then cancels the common factors. In a case where this elimination could not be efTected, the problem would become impossible with the number of leaves chosen, and it would be necessary to replace them by others. For example, let the formula be It is impossible to solve this case with two pinions o 360 X 1 ° X 1 _ a' X 3' X 5' 10 leaves, lere exists no factor 7 in the numerator which could serve to elimi- nate that of the denominator. On the other hand, if one chooses the two numbers 14 and 10, one will have 360 X 14 X 10 _. a' X 3' X 5' X 7 and will be able to cancel the factor 7. One forms, then, two groups with the figures which remain, and obtains, for example. Lessons in Horolo^. As proof, one will have correctly, > determine the r .0 X 14 138. Let the question be, now, I teeth in the wheels of a watch, whose balance should make 1 oscillations per hour. The formula (8) gives B CDiE bed let us choose for the pinions the following numbers of leaves : ithJ since the number of teeth in the escape wheel varies only wit very narrow limits, we can further replace the letter E by the figure 15, lor example. This number is, in fact, that which is very generally used for watches of medium size. One will then have tax 10X8 ■ or, on transposing the known terms, > X 15 Further simplification gives 16200 X aX4X4 = .ff CD. Resolving into prime factors, one obtains 3' X 3* X 5' = -5 C A with which one can form the following groups : X 5 = 90 -BCD. C = D = X 3" *X 3 <X 3' = 7a. The verification of the operation should give ,6.„ , '° X ;; X 7. X . X .5 . We will occupy ourselves, in the problems which follow, witfl.l the numbers of teeth to be given to the barrel and to the pinion cfl the center wheel. 139. The number of oscillations of the balance varied gready I in the earliest watches ; this figure was governed by no fixed rule I and would vary between 17000 and 18000. As these watches had] JV&ee/- Work. 89 no second hands, this number had onlj' a relative importance wiliiin these limits. In our modern timepieces, especially in watches above la lines{27 mm.)i five oscillations per second, which would be 18000 per hour, are generally adopted. In smaller pieces, this figure is increased to six oscillations per second, being 21600 per hour, with the object of diminishing the influence of jars in carrying, always very perceptible on the sinall balance with which these watches are supplied. A great many of the English watches beat 16200 oscillations, bdi^ 4j^ per second. Marine chronometers beat four cecillations per second, or 1440D per hour. 140. The problems which follow are the applications of the preceding theories and will aid in the better understanding of these various questions. We especially insist on the constant use of the foraiulas, and urge the pupils to accustom themselves to solve these questions by applying to each case the equation which suits it. The algebraic way is a sure guide which leads always to a correct solution and to exact results. In the following exercises we give numerous examples of the reliability of calculation which results "^nj the use of the simple formulas which we have just established. Problemi Relative to the Preceding Queitioni. 141. A barrel of So ieeth gears in a center pinion with 10 leaves; WW many turns will this pinion make zvhile the barrel makes z 1 Solution : We have the formula ( i ) which gives The pinion executes 8 turns while the barrel makes i ; one revo- lution of the barrel has then, in this case, a duration of 8 hours. 1142, How majiy turns -will this same pinion make ■while the 'el makes 4. f Solution ; The formula (i) further gives W ret While the barrel makes 4 turns the center pinion makes 32. inre the stop-works are placed in such a manner that the barrel ga Lessons in Horology. can execute exactly four rotations on its axis (127), the watch will, therefore, run for 32 hours, 143. A center wheel ■wilh 64. teeth gears in a third wheel pinioiK ■with 8 leaves ; the third wheel with 60 teeth gears in a pinion with 8 leaves also. How many turns wili this last pinion make during one turn of the center wheel ? Solution ; The formula (3) „_ A B " "ad gives us, ufter substituting, '^ Sirs" ~ ^ The fourth pinion will make 60 rotations during one revolution of the center wheel, therefore, during one hour. 144. What is the number of rotations which an escape pinion with y leaves will make during 12 koursy knowing that the centeft. wheel with 80 teeth gears iji a pinion of the third wheel with to- leaves, the wheel of which with J5 teeth gears in the fourth pinion. with 10 leaves ; the fourth wheel having 70 teeth ? Solution : We will use the general formula for a train of three, S«^°S=^ ... ABC from whence, after substituting values, „/'/ _ ,, V So X 75 X 7° _ 7300 turns. :'X7 145. What is the number of turns executed fy an escape pinion during J turns of the barrel, the wheel-work having the following teeth-ranges : Barrel .... ^6 teeth. Center pinion 12 leaves, Center wheel go " Third " 12 " Third " So " Fourth " 10 " Fourth " 7^ " Escape " S " Solution : Using the formula, we have „„ _ A B C D " "abed' or 146. Suppose we wish to caladate the number of oscillatums which a balance makes during one hour; the center wheel having 64 teeth, the third wheel 60, the fourth wheel ^6, the escape Wheel' Work 91 wheel i^ ; the pinions of the third and fourth wheels each 8 leaves and that of the escapement 7. Solution : The formula (8) gives ,, B C D 2 E N = 7 — -^ — , from whence ^ ^ ^ M = ^X^><f >^^X^^ ^ 14400 oscillations. 8X8X7 147. What should be the number of teeth in a fourth wheel gearing in an escape pinion^ knowing thai the pinion should muke 10 turns while the wheel makes i ? Solution : The equation (2) n a gives, after replacing n* and n by their values, 10 _ A this equation with two unknown quantities, A and a, is indefinite ; several solutions can, therefore, satisfy its demands. Replacing a successively by the numbers 6, 7, 8, 10 , we find for A the corresponding values, 60, 70, 80, icx>, , because 10 60 70 80 ICX> — = -^ = ■^— = -^ = — = etc. I 6 7 8 10 One obtains, then, the number of teeth in the wheel by multi- Paying the number of leaves chosen, by the number of rotations ^Wch the pinion should make. This is always practicable when the number of turns is a whole number. If, in place of choosing the number of leaves in the pinion, one t^kes the number of teeth in the wheel, the result may easily become fractional. The equation (2) can be written by making n = i A a ■= n'' Let A = 66, we would have for the preceding case, * solution impossible to carry out I I 92 Lessons in Horology. It is then preferable to choose, always, the number of leaves L the pinion, and to determine from these the numbers of teeth in the wheels, 148, To determine ike number of teeth in a third -wheel and th^ number of leaves in a fourth pinion combined in such a manner that ihe pinion makes 15 rotations while the wheel makes 2. Solution : One will use (2) and, after substituting 15 ^ -£ Since 15 and 3 are prime to each other and it is, therefore, impossible to simplify their relation, it is necessary, in order to avoid fractional numbers, that ,f^ be a multiple of 15 and a a multiple of 2 ; thus one could have The numbers ^5, 75, 90 will, therefore, be suitable for the wheel Ay and 6, 10, 12 for the pinion a. 149. We wish to know the number of teeth in a barrel and /fe' number of leaves in the center pinion in which it gears, so thai watch may run 8 days with 4 turns of the barrel Solution : From the equation (2) we find the value in which n' =^ 8 X 24 ^ 192 and « ^ 4 ; then. Replacing a by the numbers 6, 7, 8 and 10, successively, one 'J finds that ^ = 48 X 6 =288 ^ = 48 X 7 =336 v4 = 48 X 8 =384 ^ — 48 X 10 = 480. These solutions have the disadvantage of giving too great frl number of teeth to the barrel, for even on choosing for a, a pinion of 6 leaves, one obtains, still, a barrel with 288 teeth. In order to avoid this inconvenience, one sometimes has recourse to an intermediate pinion between the barrel and tbQ^ Wheel- Work. 93 center wheel ; the barrel would gear in this pinion and the wheel mounted on the axis of this pinion should gear in that of the center wheel. The difficulty is thus changed and becomes that of finding a place in the watch in which to put another mobile and of increas- ing the motive power a very appreciable quantity. In order to solve the problem thus arising, we will make use of formula (4) and we will have n A B ~ ab 192 4 — A B a b 48. Choosing for a 12 leaves and for ^10, one has ^ ^ = 48 X 12 X 10. Resolving the two members of the equation into their prime factors, one will obtain 2» X 3' X 5 = -4 ^. with which one could form the two groups A = 2^ X 3^ = 72 B = 2^ X 5 = 8<^. or else A = 2^ X 3 =96 As proof, one should find that 12 X 10 12 X 10* 150. Suppose we wish to determine the numbers of teeth and of leaves in the wheels and pinions forming the dial wheels. Description of this mechanism. The dial wheels are the mechanism whose object is to secure the movement of the hour hand. Since the center wheel makes one turn per hour, one fixes on the prolongation of its axis, under the dial, a second pinion, called the cannon pinion. This adjustment is made in such a manner that this cannon pinion participates in the movement of the center wheel during the ordinary running of the watch, although it is possible to give it a separate movement when one wishes to set the hands. Thus the center wheel, the cannon pinion and the minute hand have a common movement and make one turn per houi. The cannon pinion a gears in the minute wheel Lessons i A (Fig, 29), which carries a pinion ^ gearing in the hour wheel i This last wheel, usually placed on the cannon pinion, around whic it can turn freely, is the one which carries the hour hand. Tl hour wheel should, therefore, make one turn in 12 hours ; wise expressed, the cannon pinion should complete 12 rota while the hour wheel nwkis one. In the equation (4) one replaces «" by la and n by i, unknown quantities are then A B and a 6 ; one, therefore, has il = ^ B ^ and substituting for a and b the numbers 12 and 10, one will ha\ \2X 12X ^0 = A B. Resolving into prime factors, it becomes =' X 3' X 5 = ^4 J. which we can group in the following manner ; X 3* - 36 As proof, c X 5 3 6 X 40 12 X 10 ■ ^ 40. Wheel' Work. 95 These figures, 36 for the minute wheel and 40 for the hour wheel, are very often employed in practice ; one then gives 12 ieaves to the cannon pinion and 10 to the minute wheel pinion. Evidently other groups can be formed, such as these : ^ — 2» X 3 — 24 B 2* X 3 X 5 60, or A 2* 32 ^ - 3* X 5 - 45, or, again. A ^ 2 X3X5 — 30 B 2* X 3 - 48- :ie verification always gives : 24 X 60 32 X 45 30 X 48 = 12 12 X 10 12 X 10 12 X 10 In small watches or low-priced ones, a cannon pinion of 10 *^^^ves and a minute-wheel pinion of 8 leaves are often used ; this drives for the wheels : 12 X 10 X 8 = ^ i? and 2« X 3 X 5 = ^ ^. Trhe two groups ordinarily employed are : ^ = 2X3X5 = 30 B =. 2^ = 32. Sometimes, also, a cannon pinion of 14 leaves is used and a minute-wheel pinion of 8 leaves ; it then becomes 12 X 14 X 8 = ^ i? 2« X 3 X 7 = A B, from which one can make A = 2^ X 7 = 28 ^ = 2* X 3 = 48. These last two cases always give 30 X 3 2 ^ 28 X 48 ^ 10 X 8 14 X 8 15L If it were desired to make the dial wheels of a watch whose dial was divided into 24 hours, the question would not be any more complex, since one would only have to solve the equa- tion 2/^ a d = A B, 96 Lessons in Horology. Take (or example a = 2 and i r would have 24 X 12 X 10 = : A B and [rom« hence ^^ X 3" X 5 = = A B, A = 2* X 3 = 48 J B = 2* X 3 X 5 = 60. 1 ive J 152. If the same dial ought to show, by means oi two pairs of hands of different color or shape, the division of time into 24 hours and the division into 12 hours (Fi^. 30), it would be easy to use the same cannon pinion and the same minute wheel for both seta ol wheels ; one need only add a second pinion, c, fastened on top of the first, and gearing in a second hour wheel, C, loosely fitted ( the first wheel. Admitting for the first train Cannon pinion 12 leaves, Minute pinion 10 leaves, one should have for the first train Minute wheel 36 teeth. Hour wheel 40 teeth, Choosing for c 6 leaves, one will have " n X 6' from whence r - -4 X ■ ■ X 6 . ... C j5 4S, one will have con'ecdy -"-Tiff- The two minute hands are fastened on the axis of the center wted, since in both cases they should execute one turn in ao hour ; their angle of divergence once being determined, will remain permanent. 153. Calculation of the numbers of teeth in the wheels of an Kironomical clock (_seconds regulator) which should run jj days, the weight having a drop of Sjo mm. The cord unwinds from a blinder whose radius is /j 7«w. , in which is included half the thickness of the cord. This cord is supposed to run through a P>illey. Solution : Since the cord runs through a pulley, it unwinds from the cylinder a length equal to twice the descent of the weight, 'nerefore, i66o mm. On dividing this length by the circumference of ihe cylinder, a «■ r, we will obtain the number of turns executed l^J this cylinder during the descent of the weight ; therefore, I X 3.'4i6 X 15 : 17.6. ' The cylinder makes, then, 17.6 turns, while the weight (descends 830 mm. ; or, according to the data, during 33 days, Wi again, during 24 X 33 ^ 792 hours, ■ One turn of the cylinder will then be effected in 17.6 : 45 hours. Consequently, while the wheel fixed on the arbor of the cylinder (fusee wheel) makes one turn, the wheel carrying the winuie hand mu.st execute 45. One sees at once that, in order to avoid having too great a number of teeth, one should introduce an intermediate pinion a and wheel £ between the fusee and the center wheel (Fig. 31 ). : 98 Lessons in Horolagy. In order to determine the numbers of teeth in the fusee a the intermediate wheel, as well as for the pinions a and b would employ the equation (3) FiK- SI n which n" = 45 and m ^ i. If we should choose pinion 8 and 16 leaves, we will place ■15 X 18 X 16 = >4 ^. The first member separated into prime factore gives a* X 3* X 5 = -^ -fi. Wheel' Work. 99 ^Wch can be grouped in the following manner : ^ = 22 X 3* = 108 i? = 2» X 3 X 5 = 120 We would have, correctly, _ 108 X I2Q ^^~ 18 X i6 • To determine the numbers of teeth suitable for the other Mobiles, let us note, first, that since the pendulum of this regulator ^hculd beat one oscillation per second, an escape wheel with 30 ^^^th should execute one turn in a minute (71). One can then ^^sten the second hand on the prolongation of the axis of its Pinion d. The escape wheel executes then 60 turns, while the ^^nter wheel makes i, and one will have, on employing pinions ^I 12 and 10 leaves, 60 X 12 X 10 = C A "^^liich can give ^ v^ 9 vx Z> = 2* X 5 = 80. -^s proof, one will have correctly 60 = 90<_8o 12 X 10 154. If^ in place of running jj days, one desired a clock ^mnning ij months, what should be the numbers of teeth in the "^heel'Work, with the same data as that in the preceding problem f Solution : Thirteen months calculated at the rate of 30 days is ^qual to 390 days or 9360 hours. One places n^ ^ ABC n a b c * for one sees that, in order to avoid having wheels with too many teeth, a second intermediate wheel must be introduced between the fusee and the center wheel. One will then have 9360 ^ ABC , 17.6 ab c ' that is to say, while the fusee wheel makes 17.6 turns the center wheel should make 9360. Since the numerical expression 9360 17.6 cannot be employed because of the fraction in the denominator, we will transform it by means of the following operation into an ?.SV^VV>^ I loo Lessons in Horology. equivalent fraction, having as denominator a whole prime 17.6 " '^' ' multiplying this quotient by 11, one obtains the whole numbo' 5850, then 936° _ 585^ 17.6 11 ■ One will have, consequently, ^^ abc = ABC. In choosing the numbers of leaves for the pinions, care must be taken that 11 is found as factor in one of these numbers, order to be able to eliminate the denominator. Let us take, then, the figures 22, 16 and 14 ; we will have -^ X " X 16 X 14 = ^ -ff C and 3' X 3' X 5' X 7 X II X 13 ^ ^ s C. After cancellation, one could £orm the following groups ^ = 2« X 3 =192 B ^ 2 X 5 X 13 = 130 C = 3 X 5 X 7 = 105. If one found these values too great, one could choose pinions with fewer numbers; for exampie, 10, 11 and 12 leaves, and one would have, in an analogous manner, ^ = 2= X 3 X 5 = 120 5 = 2 X3 X13-- 78 C = 3 X 5' =75- For both cases we would hate correctly, 193 X 13° X 105 _ izo X 78 X 7 5 ^ 5850 22 X 16 X 14 la X II X I" II ■ For the other wheels of the train, the case is the same the preceding example. By introducing a third intermediate wheel, one could succeed in making such a clock run for 10 years, but there exist practical disadvantages which make this combination seldom used, 155. How to deiermine the number of teeth in a third wheel which has been lost, knowing that the balance should beat 18,' oscillations per hour and knowing the numbers of teeth in ike other wheels and pinions f Wheel' Work, loi Solution : Let us call x the unknown number and let the Center wheel have 80 teeth, Third wheel pinion 10 leaves Third ** ** x ** Fourth ** ** 10 ** Fourth ** ** 70 ** Escape ** ** 7 ** Escape ** " 15 ** The formula (8) admits of placing 18000= 8oX^X7oXaXi5 . 10 X 10 X 7 or, simplifying, 18000 = 240 X, and -- 18000 . .. X = — -— = 75 teeth. 240 The lost third wheel, therefore, had 75 teeth. 156. Ify in the preceding problem^ the last mobile had been the third-wheel pinion^ how would the equation be solved? Solution : We would have in an analogous manner : ,8000 = 80X75X70X2X15^ or ;ir X 10 X 7 - 180000 18000 = and 180000 18000 10 leaves. 157. Still using the preceding data, let us suppose that the pinion and the escape wheel were both lost, and let us propose to determine their teeth ranges. Solution : We will have, in this case, two unknown quantities, which we will designate by x ^nAy ; the equation (8) will be written l8ooo==^X75X7°X2^ from whence 10 X 10 X jv and 18000 = 8400 X y 18000 X On simplifying. 8400 y 15 X 7 y The wheel, then, should have 15 teeth and the pinion 7 leaves. I02 I-£ssoiis in Horology. 1S8. In the last problem, we arrived immediately at the ^^ numbers ; this does not always happen. Let it be desired, a^S^ | second example, to find the numbers of teeth and of leaves i center wheel and a third pinion whieh have been lost: Solution : One has ,«™ _ :^ X 75 X 70 X a X tS from whence one obtains y X • X 7 mplifica This result shows us that the center wheel should have eight times as many teeth as the pinion has leaves. On replacing succes- sively j/ by 6, 7, 8, 10 and 12 leaves, one will obtain the followii^ solutions : Fo V = 6 one has X = 6 X 8 = 18 teeth V = 7 X ^ 7 X 8 — Sb " y = 8 X = 8 X 8 = 64 " y = ' 12 X 1 Several solutions can, therefore, satisfy the demand, and the one which suits best must be chosen ; it is evident, here, that with rela- tion to the numbers of teeth in the other mobiles, a center whed with 80 teeth and a pinion with 10 leaves are perfectly admissible, 159. Indicator of the Spring's Development in Fusee Timepieces; Murine chronometers and a great many fusee watches carry 2 auxiliary hand placed on the dial, and with its center on a straight line between the point of XII o'clock and the middle of the center wheel. The object of this hand is to indicate, on a small dial, the number of hours which the chronometer has run since it was last wound. It gives notice, thus, of the proper time for rewinding the chronometer, an operation which then brings the hand ba(j| to zero. This mechanism, easy to establish in fusee watches, becomes di/licult to introduce in other kinds. Let us take up the first for, the present. IV/teei - Wori:. 103 When one winds one of tliese timepieces, one causes the axis oi the fusee to turn, which, once the spring is wound, takes a movement in the opposite direction. If, therefore, one places on this axis a pinion communicating its movement to the wheel on ■which the small hand is fastened, the mechanism will be complete. The indicating dial is arranged in such a way that the figure XII may be wholly preserved ; the hand cannot, tlierefore, make a com- plete turn. Let us suppose that while the watch runs 56 hours with 8 turns of the fusee, this hand may make seven-eighths of a turn and that there thus remains one-eighth of a turn, which is taken up by the lower part of the figure XII (Fig. 32). The question now, is to determine the numbers of teeth ii '^heel carrying the hand a ^"e fusee, in order to produc' given data. While the fusee executes r. " = J; therefore, {2) ; pinion fastened on the axis of . movement conforming to the S turns, the hand should make i ^The number of teeth in the wheel should, therefore, be a mul- le of 64 and the number of leaves in the pinion a multiple of 7. One can have for « ^ 7, j4 =; 64 a = 14, A = 128, etc. When the chronometer is running, the fusee is animated with a movement to the left ; the hand, therefore, turns to the right, in .the same direction as the other hands. ^thei I04 Lftsons in Horology. It may happea that, owing to the airangemenl of the calibre of the wal!ch, one could not make the pinion gear directly in the wheel ; it would be necessary in this case to place a second wheel {{curing on one side in the hjsee pinion, and on the other in the wheel carrying the hand. This last mobile will then take a move- ment to the left, and if one wished to avoid that, it would be neces- sary to arrange two intermediate wheels between the wheel A and the pinion a. Let us note that the number of teeth in this or in these inter- mediate wheels does not modify the relation existing between the movement of the wheel A and that of the pinion a. Designating by B the number of teeth in the intermediate wheel, one has a D 160. Simple Calendar Watches. By this name we designate a ' certain class of watches having an accessory mechanism by means ol which is shown, on the dial of the watch, the date, the day of the week and the name ol the month. These indications are made by means of hands fixed on the axes ol toothed wheels, performing their revolution in the length ol time desired ; that is to say, the httnd indiciiting the date jumps each day at midnight, as does also the j one indicating the day of the week. These movements are obtained *■ by means of a wheel making one rotation in twent^'-four houis^.fl TItc dwl showing! the date is ^\ided into thirty-one parts ; the hand ' iump*. therefore, for months of thirty-one days from the figure 31 to the figuiv I. II the month has only twenty-e^ht, twentj'-oine or thirty days, the hand should be set to the figure i t^* faa Ttns b utconveoient mmI b ovcrcoine in popetoal caletidais, Mick iiKclaiu»tts are. thenlore, more conyticated. CaleMfer watdMs are often made to incficate also the rtwKKKHk Aso(teHing b nUKlein dw dial for this pmpase, •irkiMi pan wccw i vrfy representations ol dwnrioas wMdi th* WQUK prtse«ts. Tite m w t etj a vk W KJMto i a v asittMAiM*^ but iwennittm; ks •saKithieMBbcraclKMScI teoJewSv. LtH »8 "WTwat wm kwlM ■[»wnr 1 «Utlt «t feoK* ^ ■rtHtJaa^A ai« eSectad <F^. ssX It* MM aafte cQfleBiKtiia* «MsBi W^f/- l^ork. 105 B and C, of 60 teeth each. These last niobiles execute, therefore, one revolution in 24 hours. The wheel B carries a pin m perpen- dicular to its plane, which comes into gear at each rotation with the ^^Ueeth ol a "star" of 31 teeth camnng the calendar hand. This r last wheel is kept in place by a jump-spring. During the day this same pin makes the phase wheel Z. jump in the same manner. This wheel, also star-shaped, contains 59 teeth ; it is in like manner kept in place by a jumper ; on its surface are represented two moons diametrically opposed. When one of the (aces disappears behind the dial at the time of the new moon, the edge of the fol- lowing is on the point of appearing in the shape of a slight crescent. A synodical revolution of the moon (interval comprised be- tween two consecutive full moons) is effected in very nearly 2g days and a half ; * two lunations require about 59 days. This is the reason why 59 teeth are given to the phase wheel. • Exuitlr, n iija. IShoun, Mmloutee. io6 lessons in Horology. \ The second wheel C also carries a pin, intended to make the wheel S, with 7 teeth, jump ; this wheel carries the hand indicating, the days of the week. The movement ol the hand which indicates the month, is most generally effected by setting it by means of an exterior push-piece. This hand is carried by a star wheel with 13 teeth, which is kept in place during one month by a jump-spring similar to those of the- other stars. It does not require, therefore, much calculation in order to de- termine the toothing of these different wheels. 161. Suppose it be desired to determine the numbers of teeth. and leaves 0/ the wheel-work in a decimal watch, desiring to pre~ serve to the balance wheel of this mechanism the same duration of oscillation as in that of an ordinary watch. Solution : A decimal watch is an instrument dividing the length of a day into twenty parts, in place of twenty-four ; therefore, thet interval included between a midnight and a midday, or a midday and the following midnight, into ten equal parts. Each of these "hours" is divided into 100 "minutes." Let us further make this a condition : this watch should run just as long as an ordinary watch (32 duodecimal hours). An ordinary watch, furnished with the customary stop-works, can run ij^ days while its barrel makes four turns ; this barrel will execute, therefore, 3 turns in a day. In a decimal watch, the center wheel should, accordingly, make 20 rotations while the barrel makes 3. While this barrel makes i turn, the center wheel will make - = 6J^ mrns. One will, therefore, have Choosing a pinion a with 1 2 leaves, a multiple ol the denomina- tor of the fraction, it will become The barrel would, therefore, have 80 teeth and the center pinion 12 leaves (Fig. 34). Wheel' Work. 107 The fourth wheel should then execute 100 turns while the ^nter wheel made i ; we will, therefore, have B C 100 =■ a b ' Choosing pinions of 8 leaves, we have 100 X 8 X 8 = -ff C. l^educing to prime factors, one obtains afterwards 2« X 5' = ^ C ^^^ could form the two groups 2* X 5 = 80 2* X 5 = 80, The center and the third wheel should, therefore, each have 80 J ^^ and should gear in pinions with 8 leaves. There now remains ^^ to determine the numbers of teeth in the fourth and escape for 100 turns 1 turn < \,w" 1 turn <— •■ 1 turn Fiff. 34 ^■^^els, as well as the number of leaves in the escape pinion ; ^^Se numbers have to fulfill the condition declared, not to alter "^^ duration of the oscillations of the balance. Let us determine, in the first place, the number of oscillations ^hich the balance would execute during one turn of the fourth ^heel. In one day this number is equal to 24 times 18,000 ; for One turn of the center wheel it should be 20 times less, and for one turn of the fourth wheel still 100 times less, which gives 24 X 18000 ^ .„ .. ,, — 216 osculations. 20 X 100 Let us admit, as is the custom, an escape wheel with 15 teeth. The number of turns which this wheel should execute while the Lessons in Horology. fourth wheel makes one, will be obtained by dividing 216 oscilla- tions by twice the number of teeth in the escape wheel, therefore, One, therefore, places 71 -T' and, on choosing for the number of leaves a example, one will have 7i X 10 = n. D — ■Ji teeth. mltiple of 5, 10, for which gives The fourth wheel could have 72 teeth and the escape pinion 10 leaves. 162. In the problem with which we have just dealt, the second hand will not divide the minute into 100 parts, since it will make ai6 little jumps during one revolution. We could still propose to divide the minute into 100 equal parts, by abandoning the condition stipulated in the first problem, of keeping for the balance the same duration of oscillations ; in place, therefore, of making it execute 2i6 oscillations, let us imagine it as making 200 of them. With an escape wheel o! 15 teeth, one arrives at ^ = 6-/i .«™. executed by the escape wheel while the fourth wheel made i. Choosing a pinion of 9 leaves, one has 6?^ X 9 ^ 60 teeth for the fourth wheel. 163. We have still to make the calculation of the wheel-work. J for the dial wheels. This problem can have two forms : the one 1 in which the hour hand should execute i turn a day, and the one.| in which it should make 2. In the second case, the minute hand will make 10 turns white, J the hour hand makes i ; one will, therefore, write A B with pinions of 12 leaves each, one will have 10 X 12 X " = ^ S; on reducing into prime factors, From whence 2' X 3' X S = ^4 ^■ ^ = 2* X 5 =40 teeth £ = 2' X 3' = 36 " Wheel' Work. 109 If the hour hand should only make one turn a day, one then has A B 20 = Taking d a b or 10 and a = 8 : 20 X 10 X 8 = A By « X 5' = ^ ^. ^°^ could then form the two groups A = B = .8 ,8 X X 5 = 40 5 = 40. ^^^ minute and hour wheels would each have 40 teeth in this case. 164. Calculation of the Numbers comprising: the Teeth-ranges ^' Uie Wheels of a Watch with Independent Second Hand. These ^^^ches, which were ^^nstructed in consid- ^'"^ble numbers some ^^Bj^ ago, generally ^^^tained two distinct ^^ins. In this system ^ Special hand is placed ^^ the center of the ^^^l and makes one ^^nip only per second ; ^^ can be arrested for ^"n indefinite time, then ^"tcirted again at will, "Without stopping the Vatch. The office of "this second train is to tlrive this independent second hand. The principle of the me- chanism is, therefore, to release, at each sec- ond, the train which brings the hand into action. For this pur- pose the last pinion of the second train car- ries on its axis an arm p.i^, 35 no lessons in Horology. called the "whip," gearing either directly in the escape pinioi the first train or in a "star" adjusted on the axis of this lattec (Fig 35)- While the whip is in contact with a leaf of the escape pinion, it has a slightly-pronounced angular movement, scarcely percept- ible on the second hand. But when the leaf of the pinion has advanced up to a certain point, the whip becomes free and rapidly makes almost a complete turn and again comes in contact with the pinion on the nest leaf. At each turn of the whip the se hand should advance one division on the dial. At each second, therefore, a leal of the pinion or a tooth of the star must present itself to receive the whip. A lever or a cylinder escapement, whose wheel advances at each vibration of the balance, half the space which separates two consecutive teeth, can serve for this purpose, ii the number d oscillations is 18,000 per hour, therefore, 5 per second. In effect, a wheel with 15 teeth produces 30 osdltations and requires, therefore, — ^= 6 secoiitls 5 to make one turn. If one causes the whip to gear directly into the pinion, the latter should have 6 leaves ; if not, it would be necessary to fasteni a star with 6 teeth, on its axis, into which the whip should be made to gear. The movement will then be effected according to the requirements. There is a remark to be made about watches provided with the detent or duplex escapements. During the vibration in which the wheel gives the impulse t the balance, this wheel advances an angle equal to that which' separates two consecutive teeth, and during the succeeding oscilla- tion it remains at rest. Owing to this fact each tooth still pro-- duces two oscillations ; but we cannot then allow the balance to make iS.ooo oscillations, because the whip should become free at the end of every five vibrations and, the figure 5 being an odd number, there would be found, every two seconds, a vibration without an impulse, during which the whip could not be released. Watches provided with either system of escapement, should, there^, fore, in order to be used as independent seconds, beat an evat number of vibrations per second : 14,400 or 21,600 per hour,, either 4 or 6 per second. IVAeei ■ IVcrk. Ill If the watch beats 14,400 vibrations, the escape wheel ad- vances two teeth at each second ; the star of the pinion should then have -^ teeth ; but, since we cannot have a half tooth, we will give 15 teeth to this piece, which will amount to the same thing. If the watch beats 21,600 vibrations, the escape wheel advances 3 teeth per second ; the star should have ^ teeth, that is, 5 or a multiple of 5. One can give to the ordinary train of the watch the numbers ■of teeth generally employed. Concerning the numbers of teeth in the second train, we remark that, since the center wheel carries on its prolonged axis the second hand, this wheel should make I turn while the whip makes 60 ; one should, therefore, have 60 = — -r-, and employing pinions with 8 and 6 leaves, II 60X8X6 = -^^; er, reducing into prime factors, 2' X 3' XS = A B. j&ouping these factors, one can have for example. A = : B = : X 3 X . X 3 The other wheels have no other condition to fulfill, except that Ae second train should run the same number of hours as the ordinary train, generally 32. The barrel, which gives motion to the train, has then also stop works with 4 teeth, and should make one turn in 8 hours ; that is while the wheel carrying the second hand makes 8 X 60 or turns. One will have, then, here CD E^ c'd e ■ ^loosing pinions of 10, 8 and 8 leaves, one has 480 X 10 X 8 X 8 = r Z? £■, 2" X3 X 5' = r/J£, nrhtch gives the three groups of factors ; I 1 1 2 lessons i 165. If the watch has a double set of dials, that is to say, if the dial is subdivided into two small dials, the hour and minute hands of which can indicate two different times, the pinion gearing in the barrel of the independent second train carries a minute hand on the extension of its axis, as does that of the center wheel in the trains generally used. A set of dial wheels is added to each train, and one thus possesses the means of making the watch indicate simulta- neously the time of two different countries. In this case the wheel which has D teeth should make one turn per hour while the pinion, which has e leaves, carrying the second hand, makes 60. The pre- ceding figures fulfill precisely this condition, since one has, correctly. D E 64 X 60 166. The arrangement of ■waX.i^^^csXi.^A fifths at quarter seconds is similar to that of the independent seconds ; but one could only construct such with an escapement whose wheel advances a half tooth at each vibration, since the star that is adjusted on the last pinion of the second train should become free at each vibration of the balance. One could not, therefore, employ in either of these systems, detent or duplex escapements. Fifths of seconds watches, should beat 18,000 oscillations and the star of the escape wheel should present a tooth at each vibration ; this star should, therefore, have twice the number of teeth that the escape wheel has. Since this last generally has 15, the star should have 30. In place of the whip, another star, with 5 teeth, is adjusted on the axis of the last pinion of the second train. Quarter seconds watches should beat 14,400 vibrations ; the star of the escape pinion should have the same number of teeth a the wheel, and the star on the last pinion of the second train should have 4 teeth. The numbers of teeth in the other wheels are the same a the independent seconds. 167. Let us remark that these systems are out of date to-day and that they are replaced by the chronographs. These mechanisms are simpler and consequently cost less ; they are based on entirely dif- ferent principles, having no connection with the kind of problems of which we treat now. 168. Required, to find the number of turns which one should give to the winding stem, on setting a watch, to make the minute , hand move once round the dial. PVAee/ ■ Work, 1 13 Solution : Given the following numbers of teeth for the wheel in action. Cannon pinion, 12 leaves Minute wheel, 30 teeth Main setting wheel, 27 teeth Small setting wheel, 18 " Sliding pinion, 16 " C^'S' 3^)- Since it is desired to know the number oi turns which the winding stem makes while the cannon pinion makes one, this cannon pinion must, therefore, be regarded as the driving wheel. The minute wheel, which is driven by the cannon pinion, drives in Fig, 8fl *ts turn the main setting wheel ; it is, therefore, a pinion with rela- *^>Cra to the cannon pinion considered as a wheel and a wheel with ■~^]ation to the setting wheel considered as a pinion. The same *^Wing takes place for the large and small setting wheels, which also ^i*ive and are driven. One should, therefore, have I ^^B The winding stem mi ^^Rum, in order that the mi ^r One sees that the nu; l^etween the cannon pinioi ■^1 the result, and that thi la X 30 X ^7 X 18 ^ 11 ^ J_ 30 X 27 X 13 X iS 16 4 ■ St, therefore, be made to execute J of a ute hand may make i turn, ibers of teeth in the intermediate wheels and the shding pinion do not influence at /-emcnt takes place as if the sliding :, moreover, pinion geared directly into the cannon pinion. Wi: hav already established this fact when dealing with problem 159. 169. Let us n(nv seek the number of turns that one should give to the winding stem to wind up a watch which has nm a day {2^ hours'). Solution : This question deals with the calculadon of the num- ber of turns which the winding pinion should make while the ratchet fastened on the barrel arbor makes 3. Admit the following numbers of teeth ; Ratchet wheel 44 teeth Crown wheel, lower side, 38 " Crown wheel 42 teeth Winding pinion 18 " 114 Lessons in Horology. One will have, in this case (Fig. 37), and in s to the preceding example I analogous manner It is generally desired to have this number as large as possible, for the reason that the effort which must be made to wind up the main- spring, being a determined mechanical work, the lorce which must be exerted to wind the watch, will be diminished by increasing the dis- tance traversed. One sees that the number n becomes greater when we increase the number of teeth in the ratchet wheel and what are called the "crown" teeth in the crown wheel, or when we diminish the other teeth in the crown wheel and those of the winding pinion. 169a. Calculation of the Ti^in in a Watch of the Roskopf Type. Watches of this kind have a simplified train, inasmuch as their bar- rel gears directly into the third wheel. The movement of the hands is produced by the gearing of a wheel A (F'g- Sy) concentric with the barrel and a cannon pinion a placed on a tenon fastened at the center of the move- ment. The wheel A, moreover, carries a pinion b, gearing in the hour wheel B. The wheels carrying the hour Wheel 'Work. 115 and minute hands are, therefore, driven directly by the barrel. Let us hirther remark that the wheel A and its pinion should be adjusted to turn easily on the barrel, in order that the hands can be set to the hours. 169 b. We first propose to calculate the numbers of oscillations of the balance in such a watch, the numbers of teeth being known. Suppose Number of teeth in the barrel C ^ 128 '' ** ** third wheel . . . Z> = 84 ** ** ** fourth ** . , , E = 60 '' ** ** escape ** . . . Z' = 15 <( (i = 8 leaves '* three pinions , . . ^ d = 7 = 6 ** ** teeth ** minute wheel . . . A =^ 72 hour '' , . , B = 66 leaves ** cannon pinion . . a = 18 minute wheel pinion . d = 22 The cannon pinion should make one rotation during an hour. We will obtain the time of one rotation of the barrel by the quotient A _ 72 _ T - Is - 4- The barrel takes four hours to execute one turn on its axis. The number of oscillations accomplished by the balance during one turn of the barrel, that is, during four hours, will be expressed by the formula ., CDE2F A N = -= , c d e and during one hour .. CD E F N = -= . 2 c d e We will have, consequently, 128 X 84 X 60 X 15 .„ ,. ^ = — 2 V 8 V 7 V 6 — "^ 14400 oscillations. The train of the dial wheels will give, properly, 72X66 _ 18 X 22 ~ "• 169 C Suppose now we wish to calculate the numbers of teeth in the train of a Roskopf style of watch, knowing that the balance should make 16,200 oscillations per hour. ii6 Lessens in Horology, Let us admit, as in the preceding case, that the barrel makes one turn in four hours. We wiD have CDEiF 16200 = J . A c d e Choosii^ pinions of 8, 7 and 6 leaves, one will have 16200 X2X8X7X6= CD EF, and on reducing the first member into prime Actors, 2» X 3* X 5* X 7 = CDEF, with which we could form the following groups : C D E F - 2« X - 2« X - 2* X - 2 X 3 3 3 3' X X X 5 — 7 — 5 — 120 teed 84 •• 60 •• 18 •• The train of the dial wheel should give A B a b 12. Choosing a 22 and b 18. one has A B — A B — 12 2* X X 22 3* X 18 X II, from whence, for example, A : B = 2« X - 2 X 3' 3 X II : 72 teeth 66 •• CHAPTER IV. Gearings. 170. Definition. The theory of gearings nas for its object the study of the transmission of the mechanical work from one wheel to another. 17L Let us suppose, at first, that we have only one wheel gearing in a pinion and that in place of the complicated force of the spring we have a weight P (Fig. 38) acting through the medium of a thin and flexible cord on a cylinder whose radius is equal to the . ^. unit and which is fastened concen- tncally to the axis of the wheel. / Let us, at the same time, admit : that the resisting force be represented t (^ hy a weight Q suspended in the same \ manner as P from a cylinder adjusted \ on the axis of the pinion and with a '**,.^ radius equal to the unit In further i imagining this system animated with ^ ^ uniform movement, the gearing will ^ perfect if, at no matter what instant of the movement, the work of the force P is equal and in the contrary Erection to the work of the force Qy the relation of the forces ^ and Q being properly established. Since these forces are in the same direction as the path tra- versed by their point of appUcation, the mechanical work effected, ^ Measured by the product of the intensity of these forces by the ^stance traversed (37). If the relation of the forces P and Q is correctly chosen, their "^§free may be arbitrary, and, consequently, they can be supposed ^ Very small or even as nothing. Therein is the basis of the ^^Portant theory explained in kinetics. 172. One can also exclude the movement and devote oneself ^^re especially to the transmission of the force. We will examine the gearings from this double point of view. U7 Flu. ^8 Il8 Lessons in Horology. 173. Practical Examination of a Gcarlne:. Let us place a'D and a pinion in a deplhing tool, in such a manner that the two movers may be sufficiently free, but witliout play between the points of the instrument. Regulate the distance between the two movers until the movement of the wheel produces that of the pinion. Impart then a rapid movement to the wheel : we will establish a good gearing if the movers conserve this motion long enough, and without any othM noise than a certain hissing sound easily recognized. The move- ment imparted should, moreover, diminish gradually and not abruptly. Let us remark that, in order that this experiment may succeed properly, the pinion should be furnished with a wheel, per- forming the office of a "fly," so that the movement may con- tinue long enough. One can also examine a gearing from this point of view by placing the movers in the watch and proceeding in the same manner. We have thus decided whether or not tha gearing transmits the movement properly ; let us now see if it transmits the force correcdy. Let us use, as in the previous case, the depthing tool, and place in the same manner the movers between the arms of the instrument. Let us then create a resisting force acting on the pinion, and, for this purpose, let us press tightly together the points between which the pinion is placed. The gearing will be found established in proper conditions if, after imparUng a mo\'emcnt to the wheel, one feels no jerks in the transmission and has only the resistance OE friction to overcome. It is necessary also to assure oneself of the "play" existii^ between the teeth of the wheel and the leaves of the pinion and <^ the proper space between the points of the teeth and the bottom dt the pinion's leaves. When the gearing is placed in the watch movement one can create a resisting force by pressing the end of a wooden peg against: the end of one of the pivots of the pinion ; on causing the wheel to- turn with the aid of another peg, one could assure oneself, as in the preceding case, of the qualities of the gearing considered. 1?4. Let us observe that when a gearing transmits the move- ment properly, it transmits equally well the force, and when one di these conditions is fulfilled the other is, also. It is, however, good, for a careful examination, to use the two methods, for certain defects make themselves felt more readily by one than by the other of the . two modes. Gearings, lig 17S. One will find that for the preceding experiments to indicate a good gearing, they must fulfill the three following conditions : 1st That the distance between the centers of rotation of the wheel and pinion must be exact 2d. That the shape of the teeth and of the leaves must conform to theoretical profiles. 3d. That the total radii of the wheel and of the pinion corres- pond to the mathematical calculation. We would study separately each of these three conditions, Jfhidi summarize all the mechanical theory of gearings. lint. — Dutance of the Centeri. 176. Primitive RadiL Let there be two wheels without teeth ©and Cy (Fig. 39), one driving the other by simple adhesion and ■without slipping. When the wheel O has turned a certain angle « while driving the wheel 0' , the point of contact a, has arrived at b, for example, the same point of the wheel O has then reached 6' in such a way that arc ab = arc a^ . since the movement is effected without slipping, H We can note, (i) I O b For two wheels having a reciprocal movement, this relation is precisely that of the angular velocities (34J : constant when these inders have a circular base. Moreover, if the wheel O Lessons in Horology. 1 has accomplished a number of rotadons c, the wheel (y has ma^ number <^ and one would have the new relation, 177. Although the transmission of mechanical work by ai| contact may not be employed in horology, at least in a dl manner, one finds, however, numerous applications in the won the practical man. In these cases the- wheels are not ordinaril contact ; a certain space separates them, and to produce the mi ment of driving one by the aid of the other, we wrap around g both either a cord, or, perhaps, a leather strap called "the bao Thus, for example, the cord of a foot-wheel or hand-wheel in watchmaker's lathe transmits the movement, it may be, to a couu shaft, or directly to a pulley mounted on the lathe ; the bow st transmits, likewise, the mechanical work produced by the hi which gives motion to it, to the pulley around which this t is wrapped. 1 1?8. Let us examine, in the first place, the case of two puj connected by a cord or band (Fig. 40). Let us first establish "^ fact that, if the two sides are not crossed, the two wheels the same direction ; if they are crossed (Fig. 41), the wheels in contrary directions. The angle a corresponding to i turn of the first pull equal to 2 » ; for w turns it is 2 ■» fi- The same for the second pulley : the angle a' is equal ts for I turn and to 2 « n' for n' turns. One can then write Gearings. Wheji n, r and r' are known, one has for fif r 121 If' = II f^* and if, as is generally the case, n is equal to i, one has simply -"f- The number of turns executed by the second pulley while the &^t makes i is then equal to the relation between the radii of the ^0 wheels. 179. Applications. An arbor makes loo turns to the minute ; a* ^furnished with a pulley whose diameter is equal to o.yo m. A band transmits its movement to a pulley of 0,4.0 m, diameter placed ^ a second arbor. One desires to know the number of turns made by the second pulley. We will have from the preceding relation nf icx) X 070 0.40 Since n = 100 ; 2 r = 0.70 and 2 r' = 0.40, then, performing the calculations, nf = 175 turns. A pulley of 0.80 m. diameter executes go turns to the minute ^ what should be the diameter of the pulley driven^ knotving that U should execute 160 turns during the same time f The formula — = — can be just as well written 2 r n IP'' Lessons in Horology. and from thence one establishes in figure and perlorming the calculations 2 r' = 0.45 m. 180. It often happens that a tool, such as a lathe, a counter sink, or a drill, should run at different speeds, in order to satisf the necessities of the work. One installs then on the driving arb< a multiple pulley, tapered pulley or speed cone. On the driven J arbor is likewise found a similar pulley, but always in the contrai manner. It is only necessary then for one cord to be placed on the different pairs of pulleys which correspond. The sum of the radii of two corresponding pulleys should then be constant. 181. Let us now suppose the case of a ioot-lathe. The cord of the large wheel is wrapped around the groove of a counter-shaft pulley and transcnits the movement to this counter-shaft. Another cord is wrapped around another groove of the same counter- shaft, but of a different radius, and transmits the movement of the arbor \ to the pulley fastened on the lathe. What is the relation between J the number of turns of the first wheel and that of the last? Let us designate in a general manner the nmwber of turns of the large wheel by . , . n " " " " " counter-shaft by , , . n' " " " " " pulley of the lathe by . n" " radius of the large wheel by J^ " " " small groove of the counter -shaft by /i' " " large " " '■ " " r ' pulley by . . r" small large We then have (Fig. 42) or, replacing n' by its value, Since, in this first case, the wheel drives the small pulley the counter-shaft, and the large pulley of the counter-shait drives 1 Gearings. 123 Ihe small pulley of ihe wheel, one obtains the greatest number of turns made by the arbor of the lathe. It is moved, then, with the greatest speed. li, on the contrary, we guide tlie cord of the large wheel in the large groove of the counter-shaft and the sec- ond cord, wrapped in the small groove of the counter- shait, into the large groove of the pulley of the lathe (Fig. 4j)> we shall obtain a lesser speed. Let us remark that, sir It IS a mechanical work which | should be transmitted, accord- "lE as the speed of the last pulley diminished, the force increases, and reciprocally. Thus, when one wishes to ton a piece of soft metal, such as brass, one arranges the cords in the manner to obtain a peat speed, on condition, always, that the object to be turned is of small dimensions. On the other hand, if one has a hard piece of nietal to turn, such as tempered steel, or an object of large diameter, " is proper to arrange the cords in such a manner as to obtain Iess speed. In the second case (Fig. 43) one has, in an analogous n |p the first p p, Fig. *« 182. Numerical ApplicaUon. Let a =; I. Ji ^ 400 mm. J?' 30 mm. r ^ 50 mm. r' =^ 20 mm. r" 40 mm. For the case of greatest speed, one will have 400 X 5° " 30X » > as to obtain 33M turns. 124 Lessojts in Horology. 183. The transmission of force by the means of wheels, or rolling cylinders driving each other by simple contact, can scarcely ever be employed in practice, because the adhesion, called ' ' torce ol friction," is very slight; the hmit being passed, slipping is produced. To obviate this inconvenience, one inserts in the wheel projec- tions, which are the teeth, gearing in the openings contrived in the pinion. One then forms what has been called the leaves of the pinion. With this arrangement the move- ment of the two toothed wheels should be made in an identical manner to that of the cylinders first con- sidered. It, therefore, follows that in a gearing one can always imagine two circum- ferences driving each other by simple contact, and ia the same conditions oE movement. These circum- ferences bear the name of primitive circumferences. 184. One calls the ^;Vi-A of the gearing the length of the arc measured on the primitive circumference of one of the wheels, extending from a point of one tooth to the similar point of the tooth which follows. The pitch of the gearing should then comprehend the space occupied by a whole and a blank of a tooth. The pitch of the gearing of the wheel should be equal to that of the pinion which it drives. Let us designate this pitch by the letter^ and call, moreover, the number of teeth in the wheel «, and the number of leaves in the pinion n'. The length of the primitive circumference of the wheel, a w r, should then be equal to ^ X «, since the pitch ought to be con- tained n times in this circumference. For the same reason the length of the primitive circumference of the pinion, 2 ' r', should be equal to^ n'. i Gearings, 125 In order to obtain a relation between the primitive radii and the numbers of teeth, let us divide the equation 2 wr = / « by 2 w r^ = / «' we will obtain 2 t t r / « . tT'k'P ~" p n' ' or, after simplifying (3) -;7 = li;- Tk primitive radii are then proportionate to the numbers of teeth. 185. Calculation of the Primitive RadiL In an exterior gearing, such as that which we have considered (Fig. 38), the distance between the centers of the two movers is equal to the sum of their primitive radii ; that is to say, one should have (4) D^^r^-r^, ^ representing this distance. Let us take up again the proportion (3) r n ^ which the radii r and r' are unknown quantities and the number of teeth n and n' known quantities. Without changing the value of an equation, one can add to each of its members the same term, or an equivalent term. We can then write r" '^ r^ ~ n' ^ n'' since the two terms -^ and -— - are both equal to i. The common denominator permits us to write r -\- r* n -\- nf r^ n' and because of (4) one y ivill also have from whence we deduce D n -^ n' (5) r^ — D *^ • In an analogous manner we would find if.\ n « n + n^' 126 Lessons in Horology. 186. rJumerical Application. A barrel ol 80 teeth should g( in a pinion with 10 leaves, what should be the primitive radii -^i the two movers, knowing that the distance tietween their cent& jxs is 11.565 mm.? Replacing in formulas (5) and (6) the letters by thi above given, one will have t" = 11.565 X s 1-565 X 1 1565 + 1.385 - 1.565. vain -^ei I These two calculations give 7' ^ 1.2 r = 10.2 As a verification, one should have D = r + r' = 10.28 187. To obtain the primitive radii, one can also simply regar"-" the distance D as divided into as many parts as there are teeth i^^ the wheel and the pinion together ; therefore, into n -)- n' parts:^ and appropriate a number n of these parts as the radius of th^* wheel and a number n' for that of the pinion. The calculation i^^ thus brought back to that of the preceding example. 188. The case of exterior gearing is the one which is most gen — erally presented in practice. In this system we will observe that the movement oi the two mobiles takes place ii* contrary directions ; whoi ' the wheel is animated witltfl a movement to the right, I the pinion will possess a I movement to the left* I 189, When the center % of rotation of the pinion is placed in the interior of the wheel' s circumference (Fig. 44), the gearings thus constructed take the name 1 . thus J name ■ darings. 127 o( inlerior gearings. In this case the pinion takes a movement in the same direction as that of the wheel. The distance between the centers is then equal to the difference between the primitive radii of the two wheels. Therefore, (7) ZJ = f- - f^. If the distance between the centers and the numbers of teeth in the wheel and pinion are known, the value of their primitive radii can be calculated in an analogous manner to that which we liave just employed to determine those of exterior gearings. We have the proportion (3), "fiich can be written ^ r^ _ n n^. r' t' ~ n' Jt' ' O"". again, r — r' _ n — fif r' «' ' on replacing r — r" by its value D, D _ a — n' r' - V ■ "Om whence we find (8) r' =^ D ^^^^ 'R an analogous manner one would arrive at the conclusion (9) ^ n ' 190, Numerical Application. Let us take as a numerical ex- ample that of a wheel with ijo teeth gearing interiorly in a pinion with 14 leaves, the distance between the centers being 8.75 mm. The application of the formulas (8) and (9) give : _ 8.75 X 14 - = 8.7s and performing the calculations, one arrives at the conclusion : 1.156 mm. : 9.906 " Tile verification should ; I ways give J 1 38 Lessons in Horology. I9L Let US now examine a kind of gearing sometimes employed and which is called rack gearing. In this case the primitive circum- ference of the wheel becomes a straight line ; its radius is, conse- quently, infinite and the number of its teeth unlimited. This gear- ing can be considered either as exterior or as interior, for the dis- tance between the centers can, equally, be D = o) + r' = a)— r' = co. To determine the primitive radius of the pinion gearing in the rack, it is sufficient for us to know the number of its teeth and the pitch of the gearing. In Fig. 45 let a b equal the pitch of one of these gearings, and place , let us call n' the number of leaves which the pinion should have ; the primitive circumference will then have for its value a w r' = v4 «', which gives (lo) r* = ^^^ 192. Numerical example. Let 2.8 mm. be the pitch of a rack gearing, the pinion must have 12 leaves, what should be its primi- tive radius? The formula (10) gives " ^ ^ 5.347 mm. A n' 2,8 X 13 =.8X 6 2 IT 2 X 3-1416 3.1416 The radius sought should then be r' = 5.347 mm. Gearings. 129 193. Application of the Theory of Primitive Radii to the Escapements. The theory of gearings finds its application not only in the wheel-work, but also every time that there is a question of tile transmission of movements of rotation around two fixed axes. It can then be applied also in special cases, such as one encounters in the study of the escapements, the mechanisms of repeaters, etc. It sometimes happens, and especially in the last cases, that one knows the distance between the centers of rotation and the relation of the angles traversed in the same time by the movers considered, and that one may have to determine their primitive radii with the object of finding out the form of the surfaces in contact. The formula ( i ) gives us the proportion ■which indicates that the primitive radii are inversely proportionate to the angles traversed in any equal times. Furthermore, one should have, when the rotations of the two movers take place in opposite directions, D ^r ^ r', and when they lake place in the same direction D = r — r". On performing identical operations to those which we have indicated (185), one will arrive at the following results ; I Ex ten or gearing, and {") Interior gearing, (13) (14) r'^n -r~r,. 194. Numerical Example. To find the primitive radii of an escape wheel and of the anchor, knowing, that while the wheel traversed an ange of 10° -- a, the anchor turns an angle of 9° =^ a!. Moreover, let the distance between the centers ht D =-- 100 mm. Let us remark that, the wheel being animated with a move- . ment to the right, the anchor possesses a movement to the left, I30 Lessons > Horology. when the tooth acts on the exit pallet, and a movement to the right when it acts on the entrance pallet. The first case is, then, that of an exterior gearing, while the second is similar to that of an interior gearing. The formulas (ii) and (12) will give us r = D — -— = 100 — ^ = 100 - a -}- a' 10 + 9 I ind ~ d 4- a' ~ '°° 10 + 9 ~ '"* 1 The formulas (13) and (14) wilt then give i. r = D ^^ _ ,,» ^'^ _ .00 > I Second— Form of the Teeth u)d Leave*. 195. General Study of the Transmission of Force in Gearings. In the chapter on motive forces, we compared the energy displayed by a motive spring to the effect produced by a weight placed at the extremity of a lever arm equal to the unit of distance, the system being in equilibrium (83). This fictitious weight F represents the moment of the force with relation to the axis around which this force exerts its action. By means of the gearing, this action is transmitted to the second axis and the problem is to find the moment F' of a force which, with relation to the second axis, would be in equilibrium with the moment F. 196. Let us suppose at first that the point of contact of the tooth of the wheel with the pinion leaf is found on the line erf centers (Fig. 46), and regard the wheel as a lever in the state of equilibrium. This system fulfills in effect all the conditions rela- tive to the lever ; the fulcrum is (9, tiie power is F; the resistance is that which arises from the wheel (7 and the moment of which we have to find. This resistence is applied at the point of contact, c, of the wheel- tooth and the pinion-leaf ; it is directed normally to the surfaces in contact ; here, perpendicularly to the line of centers and consequently following c N. It acts thus in a contrary direction to the force F. Gearings. 131 The lever arm (43) of the force N \s O c =^ r^ its moment is then ^ ^ ^ and because of the equilibrium, one should have (43) : (15) F=Nr, since the lever arm of the force F is equal to the unit On the other hand, the pinion is acted upon by two forces : one, F\ is the resisting moment to be determined ; the other, N'y Q c 'T .--• Fig. 46 coming from the tooth of the wheel O and acting, as also does the force Ny in the direction of the common normal at the point of contact. Since the pinion, as well as the wheel, is in the state of equi- librium, one should have, in an analogous manner, the equality of the moments : ,,v -,, Ar>.y ^ (16) F^ = NX r^. On dividing the equations (15) and (16) member by member, ^ Nr F' '' one has ' N' r^* The normal forces JV and iV' are equal, since their effects destroy each other ; consequentiy, one obtains simply (17) i 132 Lessons in Horology. from whence one finds the value sought (i8) F' = F~. On account of the propordoa (3) : 196a. If, for example, the moment of F is equal to 4000 gr., the number of teeth in the wheel n :^ 80 teeth and the number of leaves in the pinion n' = 10 leaves, the formula (19) would become f = 4000 8^ = 500 gr, A weight of 500 gr, suspended at the extremity of a lever arm i mm, from the center of the pinion would then make equi- librium with a weight of 400a suspended at the same distance from the center of the wheel. Let us remark at this time that if the force has diminished during its transmission, and is not more, with relation to the pinion, than the eighth part of what it wag with relation to the wheel, the speed of the last mover is, on the other hand, increased and has become eight times greater. 197. Supposing that the preceding calculation relates to the gearing of a barrel with the center pinion, let us now seek for the moment F" of the force that should be applied to the third wheel to muke equilibrium with the moment of the force of the barrel spring. We have seen, in the preceding case, that on multiplying the moment /" by the relation -", one obtains the moment of the force applied to the center wheel ; on multiplying, then, this latter value by the relation -^-, of the number of leaves in the third wheel pinion to the number of teeth in the center wheel, one will obtain the value sought, thus : («) 198. One could continue this reasoning for any number of wheels. Thus, the moment /""" that should be applied to the escape wheel to make equilibrium with the moment of the force of the spring, will be e.xpressed by _ n' tt" l»i) /^ = /■- - — J Gearings, 133 199. Let us choose as numerical example the very frequent case, The force has become 4800 times weaker but the speed of the last mover is 4800 times greater. That which, in mechanics, is lost in force is gained in speed and reciprocally. 200. We have just studied the transmission of the moment of the force from one wheel to another, admitting that the point of contact of the movers is on the line of centers. Let us now see under what condition this point of contact can be found outside of that line, in such a manner that the moment of Fiif. 47 force transmitted preserves at each instant the same value that it possessed when the contact took place on the line of centers. Otherwise expressed, the question is to form the teeth and the leaves in such a manner that the^ transmission of the force may be constant. It is necessary, therefore, that the value given by the formula (19) F' = F n' n remains the same no matter what the position of the movement. 134 LessoTU in Horology. 201. Let us suppose that the wheel-tooth and the pinion-leaj are formed in such a manner that at one instant of movement ttiis contact is found at the point c (Fig. 47), situated outsids of the line of centers. Let us find, in this position, what would be the value of the weight F' which would make equilibrium with thc' weight F, these two forces being placed at the unit of distance from the axes. The normal to the point c along which is exercised the reciprocal action of the tooth on the leaf and the leaf on the tooth, is necessarily normal both to the curve of the tooth and the form of the leaf, since these two lines are tangent at this point; it is directed along the straight line N N'. As in the preceding case, the two wheels can be compared to levers. The wheel O is, in effect, acted upon by two forces. The one, F, tending to impart to it a movement to the left ; its normal is /^ X I, therefore, F\ the other, A'^ directed in the opposite direction and arising from the pinion leaf, its lever arm being the perpendicular b, its moment is .V X Ob. Because of the equilibrium, one will have (43) (22) F^ NY. Ob. The pinion is likewise acted upon by two forces : the one, F\ whose moment is F' ; the other, arising from the pressure that the' tooth exerts on the leaf at the point c, following the normal direc- tion c N' ; it moment is N' X O' b'. Since the direction of this last force is inverse to that of F', the equilibrium is produced by the equality of the moments : {33) F' = N' X O' I/. Dividing equation (22) by (23), one has Since equilibrium exists in the system, the forces JVand .A", which have the same alignment must be equal ; in consequence, one has, after simplifying, Gearings. 135 The two triangles 6 a and O' U a are similar ; their homolo- I plus sides give ihe proportion but since (3) ~V ^ ~^' one will also have O b _ n_ (y b' n" «eretore, F _ n F' "»'"' 'roin whence one finds the value ^F' ^ F ~. 202. The value of F', identical to that which we have deter- ''^'Hed in the preceding case, is then realized, and the iorce traas- "*ltted from one wheel to another will remain constant, if the "*^l"mal common to the point of contact of the tooth and of the '^f passes, in no matter what position of the movement, through "^^e point of tangency of the primitive circumferences. 203. To recapitulate, we can deduce from the preceding demonstrations the following rule, which is the basis for the de- termination of the forms of contact of teeth and leaves. In order Ikai Ike transmission of force by gearings may remain conslani, it is necessary tkat ihe acting surfaces of the teeth-ranges be formed by suck curves that at any instant of the movement tke normal common to the point of contact passes always through the same point of tke line of centers, which is ihe point of tangency of the primitive circumferences. 204. It follows from this law that when the contact takes place on the line of centers, this point is blended with the point of tangency of the primitive circumferences. 205. Let us remark that, if the normal cuts the straight hne O 0' between the points O and (7, the gearing is exterior and the movements of the two mobiles take place in opposite directions. If the normal cuts the straight line O C outside of the points and 0\ the gearing is interior, and the movement of the two wheels takes place in the same direction. '36 Lessens in Horology. If the normal cuts the line O O* at the point Cy, the radius ^ becomes nothing and one has P' = F~ = O. the transmission of the movement of the force is impossible. If, on the contrary, the norma! cuts the line O (7 at the point O, one has in this case r ^ o and consequently : F' = F— = <xi\ the force F' becomes infinitely great, but the transmission of the movement is wholly impossible, since the primitive radius of the wheel is annulled. If, finally, the normal was parallel to the line O U, one would then have f' ^ F — = F This could be the case with the entrance pallet of the anchor escape- ment if the escape wheel should traverse the same angle a as the anchor which it drives ; one has thus (193) : the primitive radii are then infinite, 206. The law which we have formulated (203) shows us, even from the beginning, that the problem whose object is to find the curves of the teeth and leaves is susceptible of a great variety of solutions, for one may give to the teeth of one of the wheels any special form and find such a curve for the teeth of another wheel as should satisfy it, in its successive contacts with the first, according to the conditions given. However, the laws of the resistance of the materials, the wear of the rubbing surfaces, the inflexions of the J curves, are so many causes which make us, in practice, reject thw use ol a number of these solutions. 207. Let us further remark that the formula F' := F ^ is < independent of the absolute value of the primitive radii r and r" * and depends, consequentiy, only on the relation of their primitive J circumferences. Determinatioii of the Formi of Contact in G«tirin^. 208. There are several methods serving to determine the bear^fl ing surfaces of teeth and leaves ; the basis of these different con- structions rests generally on the law which we have set forth C203)J We will study here three of the principal of these. Gearings. 137 209. First— Graphic Mctliotl. Exterior Gearing. The funda- mental condition, that the common normal to the point of contact oi two forms which drive each other should invariably pass through the point of tangency of the primitive circumferences, himishes an easy graphical means to determine one of the curves, when the other is given. Let and C (Fig. 48) be the primitive circumferences of a iring and A B the given curve of the pinion in any position. 138 lessons in Horology. If from the point of tangency a we draw a normal to t curve, we will thus have the point of contact i of the leaf of t pinion and the tooth of the wheel corresponding to the posida) described. Let us remark that, in this position, the normal a i forms t same angle with the radius r' of the pinion as it does with the longation of the radius r of the wheel, since these two lines r into each other. Let us afterwards mark on each of the primitive circumference a point, b and ^, determined in such a manner that one may hav^ Through the points 6 and b' draw the radii O b and ff 6', pr( longing the first suiSciendy beyond the circumference of the wheel from the point b' trace the normal to the curve b' 2', then lay bl from the point b as summit, an angle equal to 2' b' C and mark thi point 2 making b 2 equal to b' a'. The point 2 belongs to the curve sought, for if the points b and b' arrive at the position a, the radii O b' and O b will have the same alignment and the points 2 and 2' the same position. One can thus determine as many points as one wishes, and, connecting them by a continuous cur\'e, one will obtain a form sui as /? C, possessing the ability to drive the curve A B \a such a manner that the transmission of the movement may be uniform. If one conducts the curve A B in such a way that the point Af, which belongs both to this curve and to the primitive circumference of the pinion, presents itself at the place of the point a, the point A^ which belongs to the curve sought and to the primitive circumference, of the wheel, should enter into contact with the point Af. Thence it follows that one has and also that when the contact takes place on the line of centers it is found at the point of tangency of the primitive circumferences. 210. Interior Gearings. For an interior gearing, one deter- mines the curve of contact in the same manner as for an exterior geaiiiig. One describes the primitive circumferences O and O" tangent to the point a (Fig- 49) and the curve given A B, which we will suppose anew to be that of the pinion. On drawing irom the point a I 1 Gearings. 139 fiie normal to the curve, one detennines the point of contact i cor- responding to the position given. Let us indicate afterwards on the two circumferences the equal area a b and a 1/ , a c and a (', etc., laying off from the points b, c, etc. , angles equal to the angles that the normals // 2' , tf 3', etc. , iorm with the radii 1/ O , ^ (y , etc. Afterwards making b 2^^ 1/ 2', c 3 ^ f' 3', etc. , we determine the points 2, 3, etc. , belonging to the curve sought. The only difference between this drawing and the preceding one lies in the fact that for the exterior gearing, one lays oil the angles z' b' O', 3' f' (/, etc., on the prolongation of the radii ob, Of. .etc., of the wheel, while for the interior gearing one lays these angles off from the radii themselves. Since we can choose arbitrarily one of the two curves and seek for the Other, we can see that the problem allows an infinite number of solutions ; let us remark, however, that a number among ihem preal inconveniences, and even impossibilities, for practical execution. S 211. Second— Method of the Envelopes. The centers of rd tion of the two wheels are habitually fixed and the mobiles around these points. Let us suppose, however, that a movement of rotation have been imparted to the whole system around one of the cenl that of the wheel, for example, and that this movement is execu^ in such a manner that its angular speed may be equal to, but fel contrary direction to the angular speed animating the wheel O. is evident that from this method the wheel remains in a state i repose and that the working of this gearing will remain the sait as if the two centers were fixed and the two wheels turned sini^ around their respective centers. 1 The gearing of the fourth wheel with the escape pinion | timepieces called " tourbillon " offers an example of such a mov ment. The wheel is screwed on to the plate of the watch ; movement is, therefore, null with relation to this plate. The esca pinion, pivoted in a mobile cage, turns around its center and simi taneously with the cage, whose center of rotation is also the cenl of the fourth wheel. The principle of the method of the envelopes r^ts on this w of movement. J Let us adopt, in short, any form of leaf ; on representing 1 pinion in several successive positions of its movement, arott the wheel, we will obtain the form of the tooth, on joining by tangent curve the positions that the leaf will occupy during tl movement. One can then say that the tooth is the "envelope" ofthediffert positions occupied successively by the leaf during the movement the pinion around the center of the wheel. With this method, t! tooth remains constandy in contact with the leaf, and the transm sion of the force will be effected without loss. The movement the wheel being uniform, that of the pinion will consequently al become so. U Let us take some examples : l| 212. The transverse section of the pinion leaf of a /anU ^■earing is a circle whose center is situated on the priimti circumference. Suppose we wish to determine the form of t tooth. Gearings. If (Fig. 50) the circumference passing through the points '"1 2", 3", ... is the primitive drcumference of the wheel. (y A B that of the pinion, and if this last is moved without )ing, around the primitive circumference of the wheel, the Lessons in Horology. successive centers of the pinion will occupy in turn the points o', i', 2', 3', . . . On conceiving, then, the corresponding positions of the pinion leaf whose centers should occupy the positions o, I, a, 3, . . , and on drawing the curve abed,... tangent to the leaf in these several po- sitions, one will obtain the curve of the tooth sought. One sees that. pinion leaf is reduced to a point, the form of the tooth would be an "epicycloid" whose generating circle would be the primitive circle of the pinion. If the leaf is formed by a cylindrical pin, the curve for the tooth which results from it is par- allel to this epicycloid, and is found removed a distance, equal to the radius of the pin, ■ i One could draw a sec- ^^^^^^^^T ^i ond curve tangent exteriorly ^^^^^^^^f to the several positions that ^^^^^^^H I the pin occupies during the ^^^^^^^H movement the curve thus ^^^^^^^H formed would then drive the ^^^^^^^H pin by its concavity. ^^^^^^^H The straight lines i" ^^^^^^^H ^1., which connect the ^^^^^^^^H (*' ^ points of tangency of the ^^^^^^^^H primitive circles with the ^^^^^^^^V : p point describing the epicy- ^^^^^^^^ Fie 01 cloid are normal to the curve ^ at these points. ^1 213. Fig. 51 shows that if the contact of the tooth with the ^f leaf took place at a point such as c, before the passage of the line of centers, there would be produced an abutting which, if it did not absolutely prevent transmission of the movement, « would, however, modify considerably the uniform transmission ^■^ of the force. Gearings. 143 One knows, in fact, that the normal to the point of contact should pass through the point of tangency t of the primitive circum- ferences; but one recognizes that this essential condition is not fulfilled in this case, since the normal cuts the line of centers at a point b. In place then of being (201) the moment of the force transmitted will no longer be expressed ejtcept by the value ^ Ob' friction being left out. One recognizes, thus, that in lantern gear- '"Ers, the contact of the tooth and of the leaf should commence ^ery near the line of centers.* Note. — If the normal passed through the center ff of the fnion, the movement would become impossible ; il it passed on "'e other side of O", the pinion would turn in the opposite direction ^JScily It the ) °^ tb/iooth tur i„2P« hetween Iha line of eent iPaeilou. IhB beBinnlPg "f the I pin conimences. Let X (Fig. "^asiIonBt thin Instant. The =«toiooD to tha tooth gnd to Iba jiln, pisws IShiubIi "eaia O of the plo. Tha dlsiance O J is at once llie rMios of (ho pfn and thB radiu* of curvaturo of the •Moreloid JV Oat IbB point O; let u>, Ihetefore, dosig- il at Ibirpoii tute it bj 6. The diitanca A la that wbic h, iQ the bnaalt cited Id the teit, - B Jf 8-" + S'-TTb Ij dpsinTialed by n, and the dlslaoM A X. equal to B — n, ia very nearly the diaUnce sought, be iDRle of XO with thfl line nf centern diffcra v erj Utile ftcSn a right angle (tf it diffeta -eo.lblj In the ngursit Il becmuM the radius of the pin fj Is enag^ prevanl fho confiiiloo of the lines). But th rated to J JT Bought to the radius 8 of (he pin. I L 144 Lessons in Horology. I to the movement indicated by the arrow, and the gearing would become, on that account, "interior." 214. Lantern gearing can be interior and then admits of two arrangements, according as the interior wheel or pinion carries the pins and the other the teeth, or as the large wheel carries the pins and the pinion the teeth. 215. Take, again, as a second example of the application of this construction, the straight line A B, given as the form of the pinion leaf, and let it be required to determine the curve of the wheel tooth (Fig. 53). During the movement of the primitive circumference ol the pinion around that of the wheel, the line j4 5 will occupy succes- sively the positions A' B' , A" B", A'" B'", etc. From the points ol tangency, i, 3, 3, 4, etc., let us draw respectively the perpen- diculars to these lines and through the points a, 6, c, d, thus obtained, let us make a curve pass tangent to the successive portions of the line A B ; v& will thus obt^n the form of the tooth. Let us remark that if the wheel is animated with a movement to the right, the position A"" B"" can become impossible for the transmission of the movement, for the reason that the wheel could then turn without driving the pinion. This shows us, moreover, that there exist limits beyond which the driving of the pinion by the 'I wheel becomes practically impossible. I 216. When the line A B passes through the center of the pinion, the curve of the tooth is an epicycloid produced by a ptnnt of a circle whose radius is equal to half that of the primi- tive circle of the pinion. 217. Note. — On comparing the two methods of determining the forms of contact which we have just examined, one can prove that the graphical method (209) is necessarily analogous to that of the envelopes. In short, in order to obtain the form of the tooth, we make one of the primitive circles with the given form roll around the other ; the curve sought is, therefore, in both cases, that which passes through the meeting point of the normals with j the given curve, in each of its successive positions. The reason < which has made us separate these two parts of the same whole i simply the greater clearness in the explanation of the subject 218. Let us take, for a last example, gearings formed by the j evolvent of a circle. Gearings. 145 / / ! •^ • • • • L • > I • « « t « « • ! » • \ \ I : • , « ". • • V •!* •% 4» t 1 Vig. 58 lift Lessons in Horology. The ' ' evolvent " of a curve is another curve, C C C" C" . . . produced by a point of a tangent to the first curve, whose contact changes continually, in such a manner that the distance oi the de- scribing point from the point of contact may be constantly equal to the space traversed by the point of contact oa the curve. Thus (Fig, 54). -ff' C, B" C", . . . being positions of the tangent, one should have B'C ^ B'C; B"C" = B" C, etc The curve C B' B" . . . on which the tangent rolls is the "evolute" of C C C" . . . The point C where the evolute meets its evolvent is the origin, 219. For gearing formed by the evolvent of a circle, one adopts for the motive tooth the evolvent of any circle concentric and interior to the primitive circle of one of the two wheels. The profile of the corresponding tooth for the other wheel will be determined, then, in a very simple manner. Let the evolvent Z? i? of the circle E'E' (Fig. 55) be given. In order to determine the point of contact M of this curve and ol the form sought, draw the normal A A' from the point of tangency of the primitive circumferences ; by the construction this normal is at the same time tangent to the "evolute" circled' E' . But if from the centers O and O* we draw the perpendiculars O B and (y B on this normal, we will obtain two similar triangles whose homologous sides are in the same relation. But the sides a O, a O' and Cy B' are constant, being the radii of invariable circles ; therefore, B must be also constant. Consequentiy, the normal A A' of the curve D D sought remains always at the same distance from the center O and it, therefore, envelopes a circle E E con- centric and interior to the primitive circle of the wheel and whose radius is found with that of E' E' in the same relation as those of the primitive circles themselves. The form of Ike tooth is, therefore, another evolvent of a circle. The point of contact being found at any instant on the line A A', this line is the geometrical place. Gearinf^s. i-^r Fig. 5S 148 Lessons in Horology. Thus, in gearings of this kind the place of the points of con- tact is the common tangent to both circles of consiruction. The common normal retains, therefore, a fixed position in space during the movement of the two wheels around their respec- tive centers. This right line can make any angle with the line of centers ; it is the general custom, however, to place them at 75° from each ■ other. The especial advantages of this system of gearings are, first, that the two wheels being similar and the teeth not showing any change of curvature at the passage of the line of centers, any one tooth will drive the other before as well as after the line of centers. Moreover, the construction of a wheel not depending in any way on that which it should drive, all wheels evolvents of circles can gear together ; the relation of the velocides which they have is only to be considered. This is a valuable property which allows a single motive wheel to drive at once several others, or to make several wheels gear together successively, as is the case in the screw-cutting lathe. Another advantage to be con- sidered is that the distance of the centers can vary betvreen cer- tain Hmits without the regularity of the gearing suffering in consequence. The gearing of evolvents can be interior ; the form of the teeth, in place of being convex, is then concave. This fact is an inconvenience which makes this combination little used. One can, in these cases, diminish the concavity by multiplying sufficiently the number of teeth. 220. Third— Roller Method. The principle of this method differs from the preceding, but is just as general. Let us imagine, first, any polygon, A B C D E FG (Fig, 56), tnpelled to roll without sliding the length of a line X Y. At a certain moment of the movement one of the angles. A, for example, is found in contact with the line X Y. During the rolling around this point all the points of the polygon, and with them all those which, interiorly or exteriorly, could be unalterably connected with them, describe arcs of circles around the Gearings. 149 center A. As, for example, the point H, exterior to the polygon ^tJt unalterably connected with it, will describe an arc a b during ^e instant of the rolling considered. The radii of these diverse arcs of circles will be their normals ' ^^id will necessarily pass through the point A. Let us remark that the length of each arc described depends ' ^*i that of its radius and on the number of sides that compose the F*olygon. If we suppose this geometrical figure formed with a great '^Vjraber of sides, the lengths of *^-*^e arcs described while it turns ^-*~otmd one of its sides, will *^ iminish. At the limit, that is to ^^y, when the number of sides *^^^^coraes infinite, the polygon is ^^ unfounded with a continuous ^^Tirved line, and each of the ^t^oints which compose it will de- ^^cribe, nevertheless, as the poly- ^^on rolls around an instantane- ^^us point of contact, an infinitely ^hort arc of a circle. But, how- I^ver small it may be, this arc pos- sesses, nevertheless, two extreme xadii, drawn infinitely near to each other and passing through the instantaneous center of rota- tion. Since they are drawn infi- nitely near to each other, either of these radii of curvature is, consequently, normal to the point considered of the total curve described by this point during the continual rolling of the generatrix along the line oi the directrix. This established, let there be, moreover (Fig. 57), any curve, a b c, which we cause to roll on the exterior of the primitive cir- cumference of a wheel and at the interior of that of the pinion. If, to be more clear, we suppose that a point H taken outside of this curve may be connected with it in an invariable manner, the movement of this point will be similar to that of all the points composing the given curve. During a certain period of the curve's movement at the exterior of the primitive circle of the wheel O, this point H will Si Fig. as 150 Lessons in Horology. describe a trajectory d H g ; then, when the movement takes place at the interior of the primitive circle of the pinion C, its trajectory will be the line /He. These two curves can be adopted as the- profile of conjugate teeth. In tact, we imagine that the curve ab c follows the movement of the two primitive circumferences in such a. manner that these three curves remain constantly tangent at a. The trajectories meet at //, since this point describes them both ; moreover, they are tangent there, since the normal for each is ob- tained on joining the describing point H to the point of contact a of the moving curve a b c with both of the primitive circumfer- ences established. Consequently, the common normal of the teeth, at their point ot contact, passes through the point of tangency of the primitive circumferences, and the verifica- tion of this fact suffices, we know, in order to have the curves ob- tained, adopted as forms of teeth. Let us examine from this point of view the following case 221. Flank Gearings. In or- der to obtain a profile very much used in the practice oi horology, one chooses as the generating form the circumference whose diameter is the radius of one of the primitive circles and on« takes the describing point on its circtimference (Fig. 58). In the movement of the generating circle around the primi- tive circle ol the wheel, the point A describes an arc of "epicycloid" A D. In its movement in the interior of the primitive circle of the' pinion, this same point A describes a straight line O* A, which is a radius of the circle C This plane surface ff Ais called a " fiank. Let us remark that the epicycloid which forms the profile of the tooth in flank gearings is not the same as that which we have \\ Gearings. detemined for the lantern gearings (21a). In the first case it is produced by a point o[ a circle with a radius less than one-half that of the primitive circle of the pinion, and in the second this curve is produced by a point of the primitive circumference itself. 222. We are now going to prove that in the rolling of the interior of the circle with twice ike radius, the moijing point tra- verses a diameter. If one represents in effect any position whatever, O", of the moving circle during its movement in the interior of the primitive circumference of the pinion, the angle inscribed, A' O M, has for its measure the half of the relation of the arc A' jT/ comprised between its sides to the radius }i A' O, that is to say, ^^■ One can, on the other hand, measure it as an angle to the center C by the relation of the arc comprised A' A to the radius j4' (y ; therefore, ^r^- But, if the expression of the theorem is true, that is to say, if the point A of the generating circle is carried to M along the straight line A C, the two angles A' (7 /1/and A' O A should be «qual and superpose ; we would, therefore, have the equality of the terms : The arc A' M is equal, in fact, to the arc A' A, since the rolling of the generating circle is eSected without slipping ; the two relations are, therefore, equal and the point M\^ found, in conse- quence, on the radius A 0'. Since it relates to any instant whatever of the movement, this point, therefore, does not leave the diameter O A, which is, then, properly the trajectory sought. 223. If one imagines the flank in any position whatever, as, (or instance, O' D (Fig. 59), its point of contact J/ will be obtained by erecting to it the perpendicular A M. The angle A M O being a right angle, the point M will be found on the circum- ference which has A Cy ss diameter ; consequendy, in flank gear- ings, the location of the points of contact is the generating ciraiin- ference itself. 224. An analogous reasoning to that which we have de- veloped for a preceding case (213), shows that in the simple flank gearings the driving can only take place on one side of the line of centers. 152 Lessons in Horology. 225. In order that the contact o( two similar teeth ma\' raence before the line of centers and end on the other side of that liner' ^ it suffices ii each tooth has a mixed profile formed with a flank interior" to its primitive circle ant^ with an epicycloidal part; exterior, generated by a circle with a diameter equal to the radius of the primitive circle i other wheel. Thus, for example (F'g- 60), the circle ff' furnishes i successive rollings a flank a A for the wheel O and a curve A D for the wheel O. The circle O/' furnishes in an analogous manner a flank O A ioi the wheel O and a curve A n for the wheel O. This combination : called "reciprocal" flank gearing. One can, there- fore, say that in recipro- cal flank gearings , driving takes place on both sides of the line ^ Let us add that the form of reciprocal flank gearings cannot be em- Fig. 00 ployed for interior gearings. 226> Two wheels with plane interior flanks and epicycloidal curves exterior to the primitive circles should, according to the generation of their profiles, be made especially for each other, since a wheel cannot gear regularly in several others of different diame-i ters. This inconvenience is avoided for a series of wheels that one wishes to make gear with the same wheel, by replacing in the wheds of the scries the straight flanks by curves, one chooses for general- Gearings. 153 g circle of these interior curves and of the corresponding exterior e of the particular wheel, a constant circle whose diameter dif- "s the least possible from the radii of the wheels of the series. One encounters in horology an example of this case in the gearings of the dial wheels and the setting wheels. The cannon pinioQ drives the minute wheel, in which also gears the main setting wheel ; this drives, in its turn, the small setting wheel (168). An inverse movement is produced when the hands of the watch z set to the hour, and it is then the small setting-wheel which bives the other wheels. One can, in this case, take the circle O" half of the primitive de of the cannon pinion, as generating form of the exterior jpicydoids of the wheels and aftenvards make this same generating I in the interior of each of the primitive circumferences uddered, in order to obtain the interior form of the teeth, this s then a " hypocycloid " (Fig. 61). i 154 Lessons in Horology. In practice, one substitutes very often straight lines for th^e hypocycloids, and thus one obtains a general outline recalling that of the flank gearings, although incorrect from the point of view ol its construction. 276 a. Determination of tbe Profile of a Tooth Corresponding to a Profile Chosen Arhitrarlly (according to Reuleaux l. Suppose given abed f/fT ■ ■ ■ i j ^ the profile chosen, A and B the primi- tive circumferences of the t«*o wheels whose respective centers are Oimd (/ (,K'K' 6i a). One draws the normals ag, 6^, f^. . . . jA, ji. ^j, ^k, to the' profile given, Through the points a, b. c, d . . . i, J, k. one p; arcs of circles described Irom O as center. From ,Sas center I Gearings, the lengths Sg, 6^, c^ . . . J, ^k (normals) one describes arcs of circles which will determine the intersections VI, V, IV, HI, II, I, /i, //i, ///,, IV^. This series of points, connected, form the line of the gearing (place of the points of contact). This done, from the point O as center, one describes arcs of circles passing through the points I, II, III, ... I^. 11^, III^, . . . The lengths of arcs S f, r.Z, s.j. 3.4, /.j, taken on A, will be retaken on B and will determine the lengths of the corresponding arcs ^i^. It ^x- 2i3\. 3\4i. 4ySi (instantaneous centers of rotation). If from these last one lays off the lengths of the normals a^ 6.^ ^ a b, ^1 J, =^ b 5, Ci 4i ^ c 4, etc., and if these are connected by a con- tinuous curve, one will have the profile sought, ai,i>^, c, ■ ■ ■ ii,j\ ^i- 226 b. Gearing's by the Evolvent of a Circle. In extension of that which we have said about evolvent of circle gearings (219), one can further establish, in a very simple manner, the kind of generation of the forms of contact by employing the method of the rollers (a jo). In eSect, the primitive circumferences of such 3 gearing being known and the generatrix being the straight line A A' (Fig. 55) ■ 56 Lessons in Horology. inclined on the line of centers and passing through the point of langency of the primitive circles, the rolling of this line around a tangent circle E E, interior to the primitive circumference of ihe wheel, will cause to be described by a point M of this line an evolvent D D. The rolling of the same line around a circle tangent, but interior, to the primitive circumference of the pinion, will cause to be described by the same point M a second evolvent ly D', which is the form of the conjugated tooth. U is clear that the rotation of a line tangent to a circle is effected in the same manner when, according to the condi- tion established, it must be accomplished on the exterior of the wheel's primitive circle and on the interior of the pinion's primi- tive circle. Remark : Let us further state the fact that an evolvent of a circle is nothing more than an epicycloid described by a point of a generating circle whose radius is infinite. Teedi-Rtm^e. 227. Up to the present, before approaching the details rela^ve to the distribution of the teeth on their wheels, we have been occupied solely with the determination of the curves, or prt of contact, by which these teeth mutually drive each other, without determining the points where they terminate. The time has come to pay altcntian to these questions. From the geometrical point of view, a single tooth could, stricdy speaking, suffice for the transmission of the movement ; but, in practice, there would result complications and phj-sical impossibilities, independendy of the ofistacles also very serious, arising from friction. One furnishes, therefore, the wheels with several teeth, and it is, for this reason, necessary to make them all identical. Each tooth has two profiles. Strictly speaking, the posterior face could be left any shape ; such are, for example, the teeth called "wolf," in some gearings for stem winders. It happens, however, often enough, in mechanics, that sometimes one wheel drives another, and sometimes it is driven bv the other ; therefore, the movement takes place in both directions. It is best, for this reason, to construct the two faces alike. The tooth is then "symmetrical" with relation to a radius of the primitive circle, which is, in some degree, its " bisectrix." Gearings. 157 The teeth being identical and their number a whole number, hey, therefore, divide the primitive circumference into a certain lumber of equal parts between them, which is called the "pitch" lot the gearing (,184). This pitch is subdivided into three parts, I 'Ca^fitll, the blank and the play. The full is the space measured n the primitive circle and occupied by the material of the wheel ; tlie blank is the surplus, which should remain dear to allow the iDtroduction of the conjugate tooth ; the play is an accessory blank, which does not seem at first to be necessarj- from the geometrical view-point, but which, in reality, is indispensable. Numerous causes render the play necessary for the action of EKrings : for instance, imperfections in the divisions of the wheels by the machine, the shake necessary to the pivots in their holes, the expansion of the bodies of which the mobiles are formed and the inevitable introduction of foreign bodies in the wheel teeth, are all io many reasons for this necessity. 228. It can be said that the play is the relation of the arc not occupied by the sum of the breadths of the tooth and of the leaf, to the pitch of the gearing. If we represent by p the pitch of the gearing, by a the length of the arc occupied by the tooth on the primitive circumference 3"^ by d the length occupied on this same circumference by the ^H'Kponding tooth of the other wheel, the play _;' will be expressed ''y the formula If, for e niple, we had for a givei p = 6 mm., a ^ 2.8, and b = 2.8, one would obtain the play of the gearing 6 — 1.8 — 2.8 , J - >- = -h- 229. To determine the quantity of play necessary for a gear- ing, one has to examine hvo conditions : First the solidit>' of the wheel teeth and then the space to be reserved for the free passage of small foreign bodies, such as dust or the particles which, i tably, are detached from the bearing surfaces on account of \ One or the other of these conditions can have the predominance, i according to the nature of the gearing. 158 lessons in Horology. Thus, in the gearing of stem -winding works, of the selCS gear, the rack of a repeating watch, etc., the conditions inhe to the sohdity ol the wheel teeth should evidently predominate. the gearing of the fourth wheel with the escape pinion, there 11 necessarily be reserved space for foreign bodies. The same two conditions should also guide us in the choice the fonn to give to the part of the wheel teeth which forms what called the depth of the teeth. Thus, when one desires a solid of teeth one chooses in preference the rounded depth, as Fig. tf, on the contrary, one wish^ to construct a set of teeth lea- ing place for foreign bodie one will adopt a form such s is indicated in Fig. 63 ; in tU last case, one can also use th lantern gearing. . 230. In the gearings stem -win ding mechanism, change wheels for setting, etc. , one divides the play equally betwfl the two wheels ; in the gearings of the train of thi deducts it from the pinion leaf alone, for the reason that this last mobile is made of tempered steel and offers, consequently, more resistance than the brass ol which the wheel is made. The solidity of the leaf is, furthermore, also increased by its greater transverse length. Another reason which makes us deduct the play from the leaj is that the wheels generally drive the pinions ; consequently, it i the profile of the tooth which drives the flank of the leaf after th line of centers. The curve of the tooth, therefore, must be c sufficient length to be able to drive the flank far enough to prevent as much as possible, the tooth following entering into contact befon the line of centers. This, however, is not always possible for thi pinions of low numbers. 231. For the mobiles of the train, the general rule adopted, to give in the wheel half of the pitch to the tooth and the ptedl Gearings. 159 Jalf to the blank. The pinions of 6, 7, 8, 9 and 10 leaves would pen have one-third of the pitch appropriated to the leaf and lo-thirds to the blank. In the pinions of 12 leaves and above, lone would give two-fifths of the pitch for the breadth of the leaf I and three-fifths for the blank. Thus the gearings with pinions of 10 leaves and below have a piay of - i6-i6 . - i7 - i7 c. i. and the pinions of 12 leaves and above = etc.. - A- 232. For the gearings of the change wheels, one can admit y'5 * play. 233. For those of the stem-winding mechanisms, one can be Content with ^ of play. Tturd — Total Diameten. 234. Before entering into the details relating to the determina- "On of the total diameters of the mobiles in a gearing, we Commence by the geometrical study of the curves employed in ^Orological gearings. The principal among these we know to be *l*e epicycloid. As 2 preface to this question, let us establish, first, *He theory of the cycloid. Cycloid. 235. Defmition. The cycloid is a curve described by a point *3f the circumference of a circle which rolls without slipping along a straight line. This curve is employed in the rack gearings (191), which establish a connection between a uniform transfer and a uniform rotation around an axis perpendicular to the transfer. This is then the particular case of gearings around two parallel axes in which one of the primitive circles, having its radius infinite, becomes a straight line. 236. Drawing of the Cycloid. Let it be desired to describe by points the cycloid generated by the point A oi a circle with —-diameter D (Fig. 64). One draws a straight line A A' equal to the base w D oi the I One describes the circle O with the diameter D tangent at I l6o Lessons in the point A to the line A A'. One divides the generating circum — ference and the base into the same number of equal parts, 12, foi — example, which are numbered in the manner indicated by the figure. From the point of the center one draws a straight line parallel tam the base ; this line will contain the successive places transveraed by— the center of the generating circle during its rolling. Let us indicate? on this parallel the positions of the center O, corresponding to the- positions of the circle when it is in contact with the base at the^ division points, and, from each of these centers, let us describe:- circumferences with diameters equal to that of the generating- circle. 4 Let us remark that when the generating circle has arrived at \ the center i^, the point ly of its circumference is lowered to /, and ' the point A, whose movement describes the cycloid, should be ele- vated to a height equal to the distance which the point / , is lowered. Thus, in this new position, the point A should be found on the circumference whose center is at i ^ and also on a line parallel to the base passing through i^. In the same manner, we could determine the successive positions occupied by the point A while the center of the gene- rating circlt is found at 2^,, jo. 4d> ^'^■> ^""^ ^^ connecting all the points thus obtained by a continuous line, one obtains the cycloid A B A' sought. 237. Drawing of tbe Cycloid of a Contiauons MoTement. One understands that the circle (7 is a circular plate on the circumference of which a point or pencil A is fixed (Fig, 64). If one causes the plate to turn without slipping along a straight rule whose edge coincides with A A', the point or pencil A will describe the cycloid of a continuous movement. Geari7igs. 238. Normal and Tangent to the Cycloid. Let M be any poim whatever on the cycloid A M B (Fig. 65), through which It is desired to draw a normal, then a tangent. Having traced the base A A' and its parallel E E' containing the places occupied successively by the centera of the generating circle during the rolling, we will find ■lie center O of the generating circle corresponding t o 'fie point M of the cycloid, by tracing '^om the point M "'tH an opening of •"^ compass equal *° tbe radius oi the f^n crating circle, an ^^ of a circle pass- '"§■ through the line ^ ■£'. The point intersection O will be the center of the generating circle. Dropping from the point O a perpendicular on the base A A', "*^ point P will be the momentary center of rotation of the gene- "^ting circle. Its movement is composed of a movement of trans- action parallel to the base and of a movement of rotation around '^ center O. The point P being thus the center of this combined '*iovement, the point M will describe an arc of a circle infinitely siJiall around this momentary center ; the straight line MP being the radius of this arc will consequently be the normal to the point ^ sought. The tangent being perpendicular to the normal, should pass through the point T of the generating circle ; one knows, in fact, that every angle inscribed in a semi- circumference is a right angle. 239. Evolute and Radius of Curvature of the Cycloid. The evolute A' J A of the semi-cycloid A B is a. semi-cycloid equal to its evolvent (218), Let A A' be the base (Fig, 66), B/ the axis of the cycloid generated by the point j1/ of the circumference T M P. Let us describe a circumference on the diameter/"/"^ T P \ through the point P' draw £ /^parallel to A A' ; then draw the lines M M' , _ ^ rand M' P\ On account of the equality of the angles MP T i62 lessons in Horology. and P' P M'. the two right-angled triangles PM 7" and P M' are also equal and one has M' P^ P M. from whence M M' = iP M. The straight line M M' is the radios of nirvalure of the poin M: that is to say, the radius of the circumference which has t«' i consecutive elements infinitely small, common with the curve at this point Thus the radius of curvature at any point whatever M of the cycloid is twice the portion M P cA the normal comprised between the curve and the base. Designating the angle M O Phy a, by 8 the radius of curva- ture and by r the radius of the generating circle, one has 8 = 4 r. sin i a It is thus easy to see that the foot A' , of the evolute corres* ponds with the summit B of the evolvent, while the summit of tho evolute blends with the origin of the evolvent. 240. Length of the Cycloid. The length of the portion A M*- of the cycloid A M' A\ is equal to the length of the radius C cur^-ature of the point M of the cycloid A M B \ it is equal to M M\ since this is the length of the line unwound from the cycloid portion A M' . One, therefore, has A M' = M M' = 1 M P = AT. sin J a. In order to obtain the length /' of the cycloid portion A\ M' *e have evidently the difEerence : /' = 4 ^ - 4 »■ sin ! a = 4 ^ (1 - sin 1 •). 241. Definition. The epicycloid is a curve described by a point of the circumference of a circle rolling without slipping on *»e circumference of another circle. The generating circle can either roll on the exterior or on the *Oterior of the director circle ; in the latter case the interior epicycloid '*3 called hypocycloid. We have seen that this curve is employed for the form of teeth ^ the gearing of two wheels turning around two parallel axes. 242. Drawing of the Epicycloid. This drawing is analogous to that ol the cycloid. Let us describe first from the center C the director circumference on which the generating circle O should roll. Mark on the circumference C a length A A' equal to the length of the circumference of the generating circle 0. The latter being tangent to the point A, divide its circumference and the base A A' Into an equal number of parts, 12, for example. From the center C describe afterward a circumference with a radius C O ; on this circumference will be the places occupied, iSuccessively, by the center of the generating circle ; draw then the radii C A, Cj, C,, . . . etc., prolonged to the circumference pass- ing through the center of the generating circle. Describe then Jrom the points r^, 2^, j„, . . . etc., as centers, circumferences with radii equal to the radius of the generating circle. Note now that when the center of the generating circle has arrived at ig, the point /, of its circumference is lowered to 7 ; this point has, therefore, approached C the same distance that the point A has been removed from it. On describing, therefore, from the center C a circumference passing through the division 7, of the generating circle, we obtain the point .4, by the intersection of this last circumference with that of the generating circle from point /g. m the J 164 LessoJis in Horology. In the same manner we could detennine as many poitii wished, and on connecting them by a continuous line we wouL^ obtain the epicycloid sought, as it is represented in Fig. 67. 243. Drawing of the Epicycloid of a Conttniious HoTemei C and being circular plates and A a pencil point fixed in circumference O, one understands that on making the plate O roU-J without slipping on the plate C, the pencil will trace the epicycloid J A B A' of a continuous movement. 244. To Draw a Normal, then a Tangent to the Eplcycioia.J Identical considerations to those which have enabled us to draw 4 Gearings. ■65 I tomiiJ and a tangent to the cycloid furnish us the means o[ drawing tlese same hnes to the epicycloid. If it is desired to draw a norma! ■*io the point M of the epicycloid A M B (Fig. 68), we commence "fcy seeking the center of the generating cirde belonging to the j>oiiit M of the epicy- cloid, by laying off ^^ irom this point M, the Tadius of the gene- rating circle, on the circumference around / which it is moved dur- 1 ing the rolling of its \ center. From the point Oj we figure the position of the generating circle and we find the point of tangency P, of the two circumferences. During an infi- ' nitely small period of tiovement all the points of the gene- ts ting circle, as also |those which are un- ■changeably connected v\vith them, describe infinitely small arcs of circles. Since the nor- Figros tnal of an arc of a I cirgle unites with the radius, we would have, therefore, the radius 1 MP as normal to the point M of the epicycloid A MB. On drawing afterward from the point jl/a perpendicular to the normal, we would obtain the tangent Af T, This tangent should pass through the point T of the generating circle, since every angle inscribed in a semi -circumference is a right angle. 1245. Evolute and Radius of Curvature of the Epicycioid. We have seen that the evolute of a cycloid is a cycloid equal to its evolvent and placed in such a manner that the summit of the evolute coincides with the origin of the evolvent and the summit of the \! I Lessons i evolvent is lound on the same perpendicular to the base as the of the evolute. Moreover, the two bases are parallel to each other and separated from each other the diameter of the generating cirde. As regards the epicycloid, we will see, moreover, that its evolute is a similar curve, but not equal. The summit of the evolute and the origin of the evolvent at the point A (Fig. 69) still coincide ; the summit S of the evolvent and the origin A-^" are found on the same radius £ C, but their generating circles are of different diameters. The two demi -epicycloids are contained in the same angle v4 C B which we will designate by t. If we call J? the radius of the base of the evolvent and r that of its generating circle, R" the radius of the Gearings, 167 l>ase of the evolute and t* that of its generating circle, we see that Ae length of the base of the evolvent is ^d that of the evolute R' % ^mr*. On dividing one of these equations by the other, one obtains R e _ mjr^ or i?^ e ~ w r-' R' ~ r^' On the other hand, one should still have , R — R' = 2r', "^^^yci whence one draws , R — R' r* ■■= ^^ substituting ^ R 2r nd R' R — R' R^ (I) R' = R -\^ 2r Let, for example, R = 60 mm. and r = 20 mm. , we would ^^X/e, in this case, 60 + 2 X 20 100 ^ ^nd 60-36 r' = ^— = 12 mm. 2 246. The point M' (Fig. 69) is the center of curvature of the I^C)int M of the epicycloid A M B \ it is situated on the evolute -^1" M' A, We have, in effect, arc M P = arc A P, ^ince the generating circle has rolled without slipping on the cir- cumference A A'. But, arc M P =- ra and arc A P ^ R X angle A C P Let us call the angle A C Py y and place ra = Ry, one will then have _?1 — JL 1 68 Lessons in Horology, When the generating circle with radius t* has rolled without slipping on the base B' -^i", this length of arc B' A^" is equal to ir r* . One has also ir r' = arc /* iJf^ + arc M' P\ then arc P' M' = arc P' A/', arc /* M = arc S^ P', and as ;^xz P M' = r* Y. angle P (y M\ arc B' P' = R' 7, one will then have r* X angle P (y M' = R' 7, from whence But as we have we will also have R' angle P (y M' ' iL — JL R' R' •y _ If a ~ angle P (y M" from whence a = angle P Cy M\ The point M' thus determined belongs, therefore, properly to the evolute. Since the angle M TP = M' P' /* = ^ a, and since the angles at Ma.nd at M' are right angles, the straight lines MPsind P M' will have the same alignment. 247. The straight line M M' representing the line developed is the radius of curvature of the point M of the evolvent and the length of the arc A M' developed. We have, in effect, M M' = M P -\- P M' \ or M P ^= 2 r. sin \ a and P 3P = 2 r'. sin ^ a ; therefore, M 3r :--- 2 (r + r^) sin J a. Designating the radius of curvature by 8 and replacing r' by R_^__R' _ ^ / _ R \ 2 2 V^ R -\- 2rJ' we will have (2) 8 - 2 r I I -f -=5-- ) sin , tt — 4 r -^— 1 sni -?, a. For a numerical example, let r ^^ 20 mm., ^ = 60 mm.» a 60° ; we will obtain successively. Gearings. then Log. 64 = 1. 80618 + " sin J a = 9.69897 Log. G :^ 1,50515, from whence s = 32 mm. For J^ a ^^ 90°, the radius oi curvature 8^= -^i". -5^^ 64 mm. 248. We know that in flank gearings, the radius of the gene- rating circle of the epicycloid is equal to half the primitive radius of the pinion which we will designate by r' ; one has. therefore, in this case (a), ^^ — ^ The radius R takes, then, the notation r, primitive radius of the wheel. The angle 5^ a is the angle formed by the flank of the pinion leaf and the line of centers. Under these conditions, the mula (2) becomes n and n' being the numbers of teeth (184), lj« 249. The formulas (2) and (3) show that at the origin, the radius of curvature is nothing. This fact indicates that at this point the curve is united to the primitive radius of the wheel without forming an abrupt angle. The radius of curvature increases, afterward, proportionately to the sine of the angle formed by the flank of the leaf and the line of centers ; it becomes greatest when the angle J^ a is equal to 90° ; it diminishes then to become again zero Sot j4 a- = o. that is to say, at the point of the curve's inflection. 250. Length of the Epicycloid. The length of arc of a curve is equal to the length of the line developed. Thus (Fig. 69) '(■ + ^T^)- I ^H One has most frequentiy occasion to determine the length of ^B an epicycloidal arc calculated from its origin ; it is, therefore, ^■necessary to determine its length from the point .^j". For this ^B purpose it is equally proper to take in place of the angle P P' Sf i I I Lessons n Horoiogy. its cooqilaneiitarf P' P AT, whidi we vill des^nate by | p. For the point M" the ai^le ^ f becomes thus equal to P" yt' M" irhicb obliges D9 to change the sioe ioio costDe. To obtain the length A-^" M" d the eptcydoid W," M" At let 03 remulc that this length is equal to A A^" — A M". Therefore, '-"(■+*^)-"("+*TJv)™'»- " '="(-^fl7)(--»>0; r is here the radius of the generating circle of the epicycloid A MB and R the radius of its base ; so we have (245) On substituting these two values in the above equation, we will obtain the length of the epicydoidal arc, calculated from the point of origin, thus : U) /=4^(. + ^)(.-«»4p). 251. Brst AppUcatlon. A wheel of 80 teeth can drive the leaf of a pinion with 10 leaves, after the line of centers, an angle \% = 34° 45' 48" (^57) ; what is the length of the epicydoidal arc of the tooth, starting from its origin when the primitive radius of the wheel is 10 mm.? Solution : The radius r" of the generating circle of the epicy- cloid is T^, X 10. We have, therefore, and ^ from whence one obtains *"( ^^i ) = 3.5 X 1-0625 — a.6s6a5. The natural expression of cos ^ p is it follows Then Cos i P = 0.831514, 1 — cos ip - 0.178486. log: 4-'( ■ + ^) = °t=t^' f log: (■- 05 JP ) = 0.25160 — I from whence log : / = 0.67587 — I, / = a.4741 mm. __ ^ Gearings. ^^P Remark. — The height of the ogive is equal, in this case, to T 0.42285 mm. We indicate further on, the means of calculating ■"/ "lis latter value (258). / 252. Second Application. A wheel of 60 teeth can drive the ( 'eai of pinion with 6 leaves, an angle 2 P ^ 4^° 15' 47" after the toe of centers ; what is the length of the epicycloidal arc of the ^_ tooth, the primitive radius of the wheel being 5 mm,? ^^B Solution : We have here r" ^ o. 25. Therefore, r i + ^ = 1.05 4 '^ = I ^md cos IP = 0.74006 I — COS i P =0.25994. aisequently, log: 4»' ( I + ^) = 0.021 19 + log : (i — cos i p ) = 0.41487 — I log : / ^= 0.43606 — 1 / — 0.27293 mm. Remark. — The height of the ogive is, in this case, equal to -*- 2325 mm. 253. Third Application. Similar problem for the gearing of ^ wheel of 70 teeth in a pinion with 7 leaves, the angle ^ p being ^S" 55' 15" and the primitive radius of the wheel 5 mm. Solution : We have log: 4'^( 1 + ^) = 0.021 ig. cos Jp = o.766g3 log : { r — cos J p ) ^ 0.3674 9— i i — cos ^ p = o. 23307 log: / = 0.38868 — I and / =; 0.34472 mm. Remark. — The height of the ogive is, in this case, 0.21 165 mm. The calculation is, therefore, the same for all the Hank gearings, I it is useless to follow further examples of the application, b, : I 254. The radius vector C M (Fig. 70) which we will designate by 8, forms with the initial radius vector C A ^ R, radius of the base, an angle S ; one can conceive that there should exist 172 Lessons in Horology, a relation between the radius S and the angle 6. This relation is complicated, but it has a great importance in the calculations relative to the de- . termination of the ;' total radius of the ; wheels. • If the angles Ma A = a and M C a = p are / known, we would have the propor- tion 8 « \ •••• j-^v-:-^-^^"- Fig, 70 and in the second sin a r sin P from whence 8 = r sin a sin p ' Let us now seek for a relation between the angles 6 and a, and for this purpose pro- ject the point i^on the straight line C d^; we will thus form two right- angled triangles MEa^cnAMEC In the first we have M E = r, sin a, from whence M E = {R -\- r — r. cos a,) tang. R r. sin a ^ (^ + r — r, cos a) tang. p. sin a On dividing by r, it becomes : = {—- + I — cos a^ tang. p. A A' = MA\ r a = ;? (0 + P): But therefore, P Gearings. ff^ ^ M from whence 1 ( P-:^--"' 1 then 1 (5) sin a = (^^ + 1 ^ cos « ) tang. (-^ a - 8 ). 1 255. Remark. — One can also project the point O" (Fig. 70) on the prolongation oi C M, and one thus forms the two right- angled triangles (7 H M^nA (7 H C. In the first case, we have O' H= r. sin (a + P). and in the second 0' // = (^ + r) sin p ; 11 from whence ■ 1 r. sin (a + P) = (^ 4- r ) sin p. J On replacing n by its value ^^^^1 M and on dividing by r. it becomes ^^^^H sin[A(, + s, + p] = (A + ,),i„p, ^1 which one can also write \ (., si„[A,+ (A + ,) p] = (A + .),:„ 5. 256. The calculation of the equations (5) and (6) is compli- cated ; one sees, in fact, that one can only proceed by successive approximations. We give below an example of this kind of calculation. 257. Nnmerlcal Application. To find the value of the angle . (Fig. 70) corresponding to the posidon of the point M in the epicy- cloid of the tooth of a wheel with 60 teeth gearing in a pinion with 6 leaves. We will suppose that the point M considered belongs to the point of the tooth ; it is, therefore, the extreme point of the curve of this tooth. The application of the formula (5) gives us, first, \ -ff , 60 »■ 3 - + ._-_+,_„; _ = J-_„.„s. Moreover, •-^^-"-'^ since half of a tooth should take up a quarter of the pitch of the ^ _ J Lessons in Horology. Let us suppose first, the angle .= = 80 ; we would then have P = i- '-'->• • so' = a°3o'. The calculation gives V + ^ = 21 log : + I — COSa = = 1.31861 cos 80° = 0.17365 + log; tang: p = = 8.64009 + I — COS a = 20.82635 9-95870 The logarithm of the second member of the equation (5) is, therefore, 9.95870 ; in order that equality may exist between the first and second members, it would be necessary for the above logarithm to be equal to that of sine a = sine 80°. We have log : second member ^ 9.95870 log: sin : 80° = 9- 99335 diflerence = 0.03435 The equality of the two members of the equation (5) is not verified ; the logarithm of sin a. is too great by 0.03465 ; it is sequently necessary that the value of a be greater than 80°. Let us try to take a •= 86°. We will have p = -j^ a - 9 = s° 48'. '^- + I = 21 log : -- + I — COS. a = 1.32077 — COS : 86° ^ 0.06976 log : tan : p = 8.68938 20.93024 10.01015 log : sin » ^ log sine 86° 9.99894 Difference, .01 This time, the logarithm of the first member is smaller than that of the second, which indicates that the angle of 86" is too great The angle a should, therefore, be found between 80 and 86 degrees. Let us now establish the following proportion, taking note that for 6° of arc the difference between the natural values of the cosines is 0.04586 : 0.04586 6 , . 6 X H2I ■ — - ^ — from whence :i: ^= ■■ ■ „, — , 0.01 IS I X 4586 ' on making the calculation .r = I. =4666 = i" 23'. Thus, if the difference which one obtains by the calculation of the two members were proportionate to the difference of the angles Gearings. TABLE SHOWING THE ANGLE \ M TRAVERSED BY THE PINION Of SEVERAL 08DINAXV GKARINCS DURING THE CONTACT OP A TOOTH OP THE WHEEL WITH THE LBAT OP THIS PINION. MumbracirTwilli Anglo of Driving aflerllie LlneofCenleiB Angle of DrtTing LlDeDrCBaUn D,i..^X.e Wheel 60 Pinion 6 42= >5' 17" 17° 44' 'i'f 6o° Wheel 70 Pinion 7 39° 55' 15" 11° SC/ 37.857* 51° 25' 42.857* Wheel 60 Pinion 8 37° 36' 20" r 23' 40" 45° Wheel 64 Pinion 8 37° 42' 30* 7° 17' 30" 45° Wheel 80 Pinion 8 38° o'ss" 6" 59' 5" 45° Wheel 75 Pinion 10 34° 39' 53" jO ^„, ^„ 36° Wheel 80 Pinion 10 34° 45' 48* I- 14' .2" 36° Wheel 90 Pinion 12 32° 27' 30" 32° 27' 30* in place of the 30° necessary Wheel 96 Pinion I a 3^° 33' -4' . 32° 33' 14'' in place of the 30° necessary Wheel uo Pinion IS 33° 50' in place of the 30" necessary 176 Lessons in Horology. chosen, which is not entirely the case, the angle of S6° would be i" 28' too great, which gives a new value for a, say, 86° — 1° 28'^ 84° 32'. Let us begin again, therefore, the verification for this last value. One has a ^= 84° 32' and consequently p = 2° 43' 36" |-i — 31 log:— -+i — cosa= 1.32024 ^ 0.09527 20.90473 -(- log : tang : p ^ log : sin a = Difference, 67784 One sees that we have very nearly approximated the real value of a ; if it Is desired, one can approach it still nearer by a new approximation and one arrives at length at a = 84" 31' 34" and p = 2" 43' 34". Calculatian of the Total Rsdhn of the Wheel. 258. Knowing, now, the two angles a and p, it becomes easy to calculate the value of the total radius of the wheel ; that is to say, of the radius s C^ig- 1°)> ending at the point of the ogive of the tooth. (See preceding calculation : 257. ) Making this calculadon for the primitive radius R = 1, would have : = 0.05 R for the data of the preceding calculation. One should then have (254) ^ ^ °"5 St and log : sin : a — 9.99801 log : sin : p — 8.67728 1.32073 -|- log : 0.05 ^ 0.69897 — 2 log : S ^^ 0.01970, from whence G = 1.0464. I Form of the Exceii of the I^nion Leaf in a Flank Gearing. 259. The gearings of the wheels in a watch always turn in the same direction,* and it is the teeth of the wheels that drive the pinion leaves. • ET(M!pl. howpter. Oie setting wheels, which ore driven by tUt nim. In Ihia mecliaulani. the minute Bheel does not drive the cam nioment when this Iraia ii lurnad \ij the band in setting the watch t Gearings. 177 The equation (5) can be written under the following form, remarking, however, that the primitive radii are proportionate to the number of teeth and that, in this formula, the radius r of the generating circle is equal to half of the primitive radius of the pinion. sin a = (-^ + I - COS. a ) tang. (-^ « - 6 ). When in a given gearing one finds through the calculation of the above equation, the ang i*> 360° the tooth of the wheel may drive the pinion leaf a sufScient quan- tity after the line of center, so that the contact of the following tooth may commence on this line. Such is, for example, the case of a wheel with 96 teeth gearing in a pinion of 12 leaves. We have, in fact, in this case, ka = 65° 6' and J a = 32" 33'. The tooth can, therefore, drive the leaf an angle of 32° 33', On the other hand, we see that the angle which separates two con- secutive flanks is T? = 3°-- Consequendy, one can prove that the tooth drives the leaf 2° 33' farther than is absolutely necessary. On the contrary, for the gearing of a 60-tooth wheel in a 6-leaf pinion, one has • = 84° 31' 34", from whence -^ a ^ 42° 15' 47". Moreover, I -^- ^ft Since the leaf should be driven by the tooth an angle of 60° "and it is driven in reality only an angle of 42° 15' 47" after the line of centers, the contact must necessarily commence 17° 44' 13" before this hne, since 42° 15' 47" + 17° 44' 13" — 60°, The gearing should be arranged, in this case, in such a manner that the tooth enters into contact with the leaf before the line of centers, and we have seen that, for this purpose, the leaf of the pinion must be terminated with an epicycloidal form susceptible of being driven by the flank of the tooth up to the moment when the contact is made on the line of centers ; starting from this point, t is the curve of the tooth which drives the flank of the leaf (225). 1 78 Lessons in Horology. The epicycloid of the leal should be described by a point oi » generating circumference whose diameter is equal to the primitivi radius of the wheel. 260. Since the angle at which the tooth should enter into cow tact with the leal before the line of centers is never very consider-, able, it is very rarely necessary that the excess of the tooth be per- fectly ogival. This shape would be, moreover, rather hurtful thm useful, because the friction would be increased and it would also necesitate longer teeth for the wheel, in order to allow the Era introduction of the pinion leaf in the space which separates twt teeth of the wheel. One preserves, therefore, only that part of the epicyclcA which is directly useful. In large mechanics, one simply remove! the desired quantity from the points of the teeth.* In horology, one terminates the pinion leaf by a rounde; form, an arc of an ellipse, for example. This shape of the excest should be determined in such a manner that its radius of curva- ture at the point of connection with the curve of the epicycloid, should be the same for the two lines (Fig. 71). It is evideB that, for security, one makes the contact oi the tooth and the lea commence some degrees sooner than is necessary. Thus, in thi preceding case, one will admit a contact commencing 20° before th^ line of centers, rather than 17° 44' 13". 26l> Practically, in a great number of ordinary pinions, o finds the excess of the leaf terminated by a half circle. We 2 really in a position to recognize that this form is defective, 1 cially lor pinions of low numbers. It is easy to prove that widi this shape, the tooth drives the leaf too far after the line of centers, that the point of contact does not rest on the circumference of th« generating circle and that there can be established, before the line of centers, a contact of the flank of a tooth with the rounded part of the leaf. There results from this last act a butting often m pronounced. It is probably this butting observed by horologists, which frequendy makes them exaggerate the importance of the friction of the gearings before the line of centers and which has made them believe that this last is much more considerable than that which | is produced after this line. The friction observed should only be! in reality the butdng, and it suffices to suppress this to per- J •Tbli openUou U called tbo chamferiDg of tbo (eelb. ^k 0' '79 cepiibly diminish the difficulty oi the driving. The calculation proves, in fact, that the differ- ence between what is called the "entering" friction and the "leaving" friction is not great enough to be so easily found. 262. In order to determine, geometrically, the form of the excess of a pinion leaf, it is, thereiore, established that this form should be composed of an epieydoidal arc sufiiciendy long 'o be able to enter into contact with the flank of a tooth of the *heel several degrees before the point of the succeeding tooth leaves the circumference of the generating circle. Two teeth can, therefore, drive at once two leaves through a small arc. The symmetrical epicycloids of the leaf are afterward termi- nated by an elliptical shape con- necting them. Suppose, then, it \\% i .■^ be desired to determine this curve, iiSc which has no other condition to fulfill except that its radius of Wi ^ curvature at the connecdng point must be the game as that of the ' m same point of the epicycloid. 263. Let us determine, first. \\ ■ the radius of curvature of the epicycloid at a point correspond- ing to a driving of 20° before the line of centers for a wheel with \l 1 60 teeth gearing in a pinion with 6 leaves. V 1 Since the generating form of the epicycloid of the leaf is I0 — ^ I So Lessons in Horology. a circle with diameter equal to tbe primitive radius of Ui« wbed, ' the formula (248) should be written here Admitting »^ r = 60, r' = 6 consequently 1 • = 60 = »°. we would then have 6 = 60 ( 1 + -^) sin a" = 65.454 X sin a" and, making the calculation 8 = 2.28433. 264. Let us now seek the value of the angle w, formed by the radius of curvature and the straight line passing through the middle of the leaf (Fig.71). The angle a c (? is a right angle and we have, therefore, I £ = a" ; The angle a O £, oz it follows that We have, moreover, the angle 4 since in this case the width of one leaf is equal to one-third of the pitch of the gearing (231). We will have at length w — cds^Cyda — 180= — (Tac — aCyd 265. We have afterward to determine the straight line C/ e, joining the center of the pinion to the first point of contact. In the triangle G c O' we know the sides and the angle C 0'= ir+r' Gc^\r Gearings. i8i We will have, consequently, the value of the angle a O c ex- pressed by J ^ sin ^o tang: a(yc= ' ^ i r + r^ — i r cos 4* or 1 I 2 t sin 4° tang : a (y c = . r^ o ^ + I -T— — cos 4®. The calculation gives a (7/ ^ = 190 o' 46/^ The side C c oi the triangle O c G will be at last given us by the formula Sin a (y c from whence, after making the calculation, (y c = 6.4236. 266. Let us further project the point c on e, on the straight line (y Sf designating by y the right line c e and let us determine this line. We will have y ^=^ (y c ^\n c (y d, but ^ (T £/ = a (T ^r — a (T' ^ = i9*» c/ 46''' — io*» = 9® o' 46^^; therefore, ^ _ ^^^^^ ^ ^.^ ^o ^ ^g,, and J/ ^ 1.00626. 267. We now know the radius of curvature 8 at the point c of the epicycloid, the inclination of this radius to the line passing through the middle of the leaf, therefore the angle w and, finally, the ordinate y corresponding to this same data. There only re- mains, now, to determine an elliptical curve capable of satisfying these conditions. 268. Radius of Curvature of an Ellipse. The equation of this curve being one obtains dy ^ _ b* X , d X a* y and d*y ^ _ b"- dx* a* y* ' The general equation of the radius being 8 = (■+4^)' d* y d X* 1 82 Lessons in Horology. we will obtain successively ^* X* \* • ==* A« 2LJ— ^* a«y a« J/' therefore consequently We have the length of the normal M N (Fig. 72) expres and Flff. 79 >/.+(ify-W'+'^ M N = * X* a" y iif ^ One can, therefore, place 8 = a^ M N^ we have, on the other hand, therefore S = sm a/ a*y ^* sin' w ' Gearings. 183 fron:x whence y In order to determine the value of b, let us remark that p N= — ^ — tang w * P -A^ being the sub-normal, the general equation of which is e d y b* X b* X -^ d X -^ (^ y a* * froxn whence y __ b* X tang.ze; a* a* y b* tang w ' Substituting the values of x and of a' in the general equation a*y -f b^ x*=^ a*b\ ^^^^ch we can write under the form ^'V^ewill have X* g* y* S sin* w S sin w cos* w «^ ^ b^ tang * w ~^ y tang * w y ' *^^m whence -___.,/ S sin w, cos * 2g/ \ and (2) b=^j ^ y \ y — hsmw, cos * w • The equation (i) gives, moreover, (3) a = «• ^5iJ!ali. These two last values are those of the semi-axes of the ellipse, which fulfills the conditions sought 269. Numerical Application to fhe Preceding Example. We will have y = 1.00626. w — 78®. 8 = 2.28433 ; consequently * = -%/ ^^ 1.00626' ^ j^^ \ 1.00626 — 2.28433 sin 78® cos* 78® and , a= 1.05834 • JI^^^^ = 1.62245. i84 Lessons in Horology. 270. Total Radius of Ibe Pinion. The total length of t pinion leai is composed of the sum of the two lengths C e a 'P (Fig. 71 )■ The distance e p is equal to a — x (Fig. 72). We have T.6aJ45'X 0.50266, and ■05834" X tang 7 a — jc = 1.62245 — 0.50266 = 1.11979- On the other hand, we have (Fig. 71) we know (265) and the angle (a66) therefore O' . = a c r C e O' c = 6.4236 e O' « = 9° o' 46"; 6.4236. cos g" t/ 46"= 6.3444. The total radius of the pinion will be C e = 6.3444 + (a-x)^ 1.1198 R' = 7.4642. 2?1. Note. — If the excess of the above pinion leaf we by a semi -circumference, we would obtain its total radius ii more simple manner. The primitive radius being 6, the length of the circumference is as follows :..,.., ., , primitive arcumlerence = 2 « X 6. The pitch of the gearing should be equal in length to 2 « the number of leaves is 6. The width of a leaf, reckoned on t circumference, is one-third of the pitch, therefore width of : finally, the height of the excess leaf = =- ; 3 s half of this last value, therefoi We will have, consequently, the total radius, R'=r' + -^ = S+ 1.0472 = 7.0472 or ^ in round numbers. ~ r ■ Graphical Comtruction of Gearing*. I 272. Let us suppose that we know the distance between the I centers D, also the numbers of teeth of the two mobiles, and let these be, for example, D = 240 mm. ; w = 70 teeth and «' = 7 leaves. The formulas (.185) give us the value ol the primitive radii of the a(id r" — 34a - 70 + 7 ■>■ the calculations made, one obtains r = ai8.i3 .... and ^ = 21.818 .... From the centers C and C (Plate I), previously determined, "■^e describes the two primitive circles calculated, tangent at the P'=»int a Let us suppose that we wish to construct a flank gearing, one '^n commence by determining the shape of the tooth. Describe, ^f this purpose, the generating cii'cle O" with radius equal to half *^ac of the primitive circle of the pinion, and by its rolling around *>e primitive circle of the wheel, let us make it describe the ^Iiicycloid A B. Let us remark that for the clearness of the plan, instead of Putting the origin of the curve on the line of centers, as would be the logical way to do, we have carried this construction back to another point, which, practically, amounts to the same thing. In order to construct the epicycloid, one can follow the method We have indicated (242), or proceed more simply by carrying back on several successive positions of the generating circle, such as I, 2, 3, ... , certain lengths i, I = i,o, 2, II = oi, + i, 2;, 3, III ^ 01, + ij 2, + 2, 3,, etc. On connecting the points I, II, III, .... by a continuous curve, one obtains the epicycloid of the tooth. This method, which is not exactly correct, since it substitutes the lengths of chords for the lengths of arcs, is, however, admissible for drawings in which the successive positions of the generating circle are relatively close together. We divide, afterwards, one of the primitive circumferences, that of the pinion, for example, into as many parts as that wheel I86 lessons in Horology. should have teeth. We can commence this division at any point w\ the circumference, but generally it is commenced on the line of centers, or at the point where the leaf of the pinion should be found at the moment of the last contact with the tooth. In order to determine this last position, one can make use of the table (257), which gives us, (or the gearings most generally used in horology, the angle of driving of the leaf by the tooth after the line of centers. Thus, for the gearings of a wheel of 70 teeth with a pinion of 7 leaves, we see that the tooth drives the leaf 39° 55' 15" after the line of centers. If one wishes to determine this position, graphically correct, a previous division of the primitive circle of the pinion gives the "pitch of the gearing." One lays oS this length from the origin of the cun'e A, to C ; one divides the pitch into two equal parts, one of which should be occupied by the whole of the tooth, the other by the space (231). The extremity D of the Ogive of the tooth will be determined' by drawing the prolonged radius O D passing through the middle of the tooth. From the center O, one describes an arc of a circle passing through the point D and one thus obtains the point d, extreme position of the contact of the tooth with the leaf (223). One draws, afterwards, the straight line (7 d prolonged to the primitive circumference of the pinion. The point i is then the point of departure for the definite division of the pinion. If one has proceeded with exacti- tude, the angle a O' i should be, in this case, equal to 39° 55' r One can thus prove that the first contact of the tooth with the lea!' I should take place before the line of centers and at an angular dis- \ lance from this hne of at least 11° 30' 27". The excess of the leaf should be formed with an epicycloid. arc, as we have already indicated {^^g). This epicycloid is th^J which is described by a point of a circumference with radius equal J to half the primitive radius of the wheel. The curves of the teeth being thus formed and their posidons.l determined, it becomes easy to construct the gearing, by remarkin that for the gearings of watch trains, the leaf of the pinion occupies the third of the pitch for pinions of 10 leaves and less, and t fifths for those of 12 leaves and more (231). One then limits the epicycloid of the leaf, according to what wsJ liuvc said, by conserving to it only a length sufficient for the drivin til conmience a tittle before the point where the first theoretical Gearings. OF 7 LEAVES I l88 lessons in Horology. coatact should be effected, normally. During a very short instant, two consecutive teeth are thus simultaneously in contact, and this fact suffices to insure the correct action of the gearing. The leaf is afterwards terminated by an arc of an ellipse whose. radius of curvature at the junction of the two curves is the same aa that of the epicycloid determined. One then limits the length oi the flanks of the leaves and teeth by arcs of circles with radius sufficient to allow not only the free introduction of the teeth and the leaves in the corresponding spaces, but also reserving the place which foreign bodies would occupy, dust and other matters which are invariably introduced, with time, into the sets of teeth. The gearing is thus constructed and having made the drawing on a suflidentiy enlarged scale, one could deduce from it all the rela- tive dimensions for its practical construction, as we will see later on. 2?3.* Plate II represents the same drawing to a still more greatly enlarged scale; the distance of the centers is 3200 mm., the primitive radius of the wheel 2 meters and that of the pinion 200 mm, This design allows us to show more clearly the manner in which the contact of the tooth with the leaf is effected before the line of centers; the shape of the leaf, represented in dotted lines, is semi-circular ; one sees thus that in this case the normal at the point of contact does not pass through the point of tangency of the primitive circumferences, as is the case for the semi -elliptical shape, and consequently the force transmitted has not the value whici we determined (195 and the following) h e and that there should be produced a ' ' butting. ' ' 274. The drawing of gearings of ratchet wheels, setting wheds' and dial wheels, etc. , is executed in an analogous manner examine later on the several modifications admitted for such wheel teeth. In this construction there must also be taken into considera- tion the manner in which the " play " is distributed (232, 233). Pnctic&l Application of the Theoiy of Gearingi. 275, In practice there are presented problems of different natures in which it is desired to determine the relative dimen- sions of wheels and pinions. ■A« Plnlp II has, for lark of sniM. benn re{iuMi3 oiip-hnlf, Ibp rtlstanee of Ihe cei IIW mm.; lh« primitive radius of llie wheel. I m., end Uiat of tlie piuiou 100 mm. "1 :it ■ igo Lessons in Horology. It is evidenl that, at first sight, the use of a suitable instni-^ ment to establish these sizes becomes very important to the work-- man, for the reason that it saves him all calculation. We will cites the one which is the most exact and at the same time the mostf simple to use. -^^~> y:^l i; g;^— -t jft 276. The Proportional ■ Compass and Its Use. The pro- 1 portional compass, in its most I rational arrangement, is formed I of two rule plates, straight, and -I divided into equal parts ; they are J joined together at one ol thei tremities by means of a hinge i (Fig- 73)- These rules can be fastened iaj any position by means of a damping'l screw V. Their divisions should be ex- actly corresponding, equal to each other and numbered. The point ; found at the hinge, summit of the angle b O b'. The proportional compass is based on the fundamental principle of similar triangles, in which the homologous sides are proportional. Thus, imagining the primitive diameters of a wheel placed at the division of the compass correspond- ing to its number of teeth and the primitive diameter of the pinion at the division corresponding to its number of leaves, one should have the proportion b b' o b rig. 73 Since, in a gearing the number of teeth of the mobiles should be to each other as their radii, or their primitive diameters, one understands that to determine the primitive diameter of a pinion, knowing that of the wheel, it suffices to place the latter at the division corresponding to its number of teeth and, for this purpose, to open the two arms of the compass the proper distance. The Gearings, 191 primitive diameter ol the pinioD should then ccMncide with the (^vision which corre^xxids to its number of lea\'es. The proportion ^ then found to be verified. But, as has been shown before, we run against the difficulty of 'lot being aUe conveniendy to fit the primitive diameters of the two Jiiobiles in the ccmipass, since these diameters are only thecH^caL The difficulty has been overcome in the following way : 277. On dividing the primitive diameter of any wheel by the dumber of its teeth, we obtain a length which we call ^^dietmetrical Pitch^^ of the gearii^. The proportional compass always gives ^c diametrical pitch by its di\asion j^ ^hen the wheel is placed so that ..•-*'* *'• ... ^^ primitive diameter corre^x>nds in ^*^e instrument to the divisicm of the dumber of its teeth. • V^ But, if we measure the height Vji -J... - ?V..g. .. ^f the ogive a b (Fig. 74) and, on \J^ Account of the one which is oppo- site, we douUe this value, if we after- ^2xA divide this figure by the dia- '"^ -^... .. •••' metrical pitch, we obtain a quotient Pi^. 74 which, added to the number of teeth, will give the total diameter of the wheel in units of diametrical pitch. This diameter is then 2 a b d being the pitch considered. On now placing the total diameter of the wheel at the diN-ision . 1 a b its primitive radius will be by this fact placed at the division n. The same for the pinion. 278. Example. Let us consider a wheel with 60 teeth gearing in a pinion with 6 leaves, and let us represent graphically this wheel with a primitive radius of 540 mm. The diametrical pitch should be 60 192 Lessons in Horology. Let us describe the epicycloid of the tooth by making a gene- rating circle with radius equal to haJf the primitive radius of the pinion, roll around the primitive circle of the wheel ; r" being the radius of the pinion. One will have 1'" -- So x'a Let us now calculate the length of the chord c, which subtends the half of the arc occupied by one tooth. We have the formula and One will have, therefore, (^ = 3 X 540 X sin 0° 45' log r 1080 = 30334238 log sin 0° 45' = 8.1169262 log: (1080 sin 0° 45') = 1. 1503500 Consequently, one will have Chord of one-quarier of the pitch = 14.1367 Let us lay off this length of / on a (Fig. 75) and draw the radius a a prolonged to the point i belonging to the epicycloid of the tooth; o 6 is then the total radius of the wheel and a b the height of the ogive of the tooth. On measuring a 6, we will find it equal to 25 mm. and we will have the total radius of the wheel expressed in units of diametrical pitch, by the s 4 60 + OOS- 60 + 2.77 = 62.77. One will place, therefore, the total diameter of the wheel at the division 62.77 of ^he compass, so that its primitive radius corres-' ponds to the division 60. 2T9. One could proceed in an analogous manner for the pinion. Let us remark, however, that while the height of the ogive of the wheel is fi.ted, since it is formed by an epicycloid described by a point of a generating circumference with given radius, the excess of the pinion leaf is not so easily determined. • Ona conid have obtBlned 1hl9 result without ths aid at UisiaotaeXiy, hj noUnf the ircf and chords nr emal) snglu dlBer very litUB from eiuh other. Gearings. The form of the excess which one finds in a very great num- ber of pinions is that of a semi -circumference with radius equal to half the breadth of a leaf measured on the primitive circumference. This form, although we know it to he t>ad, especially for pinions of low numbers, ofTers, however, a ready "leans for the calculation. / \ Suppose n' to be the number of leaves in the pinion. The primitive diameter expressed '" function of the diametrical pitch ™1 be likewise w', since it is divided ,- "ito as many equal parts as the pinion liis leaves. The primitive circumference is, I^^erefore, circumference ^ ir r/ ^d the pitch of the gearing I If this pitch comprises a third Tor the full tooth and two-thirds for the space, the length of the arc cor- responding to the thickness of one leaf being double the radius E of the ^ftdrcle of the excess, one will have fis.'ts ■ '-^T ^B There must, therefore, be added a value equal to s to the two ^^wctremities of the primitive diameter of the pinion expressed in ^Mlnits of diametrical pitch : ^H Total radius =^«' -\- — = «' -|- 1.05. m ^B^ Thus, for the pinions whose full pari of the pitch is equal to half of the space and whose excess has the form of a semi-circle, the total diameter should be stopped at the division corresponding to the number of leaves increased by 1.05. kFor the gearing which we will consider, of a wheel with 60 teeth d a pinion of 6 leaves, one should place the wheel at the division .77 and the pinion at the division 7.05. H)4 Lessofis in Horology. 280. Let us again make the calculation for a pinion in which the leaf is two-fifths of the pitch (12 leaves and above). As in the preceding case, the pitch of the gearing is equal to « and the radius of the ogive , _ * . " " "s" ■ as 3 8 must be added to the primitive diameter, we will have ' ' - f - "=■ Tlic total diameter expressed in units of diametrical pitch is, therefore. 281. In the case of pinions with the excess semi- elliptical, the height of this excess becomes superior to those with which the pre- ceding calculations have furnished us. Wc have seen that the calculation of this value is complicated (859 and the succeeding) ; therefore, without entering into other details, we refer, (or these values, to the table which we give further on (283, third column). Thus, taking up again our gearing of a 60-tooth wheel and 6-leaf pinion, we find in this table, that the total diameter of the pinion expressed in imits of diametrical pitch is 7.4648 ; that is to say, 7.5 in round numbers. 282. Practically, to employ in a proper manner the proportional compass, one must, therefore, commence by examining the excess of the pinion leaves, estimating it with relation to the breadth of the leaf. It one judges, for example, that it is equal to half the thick- ness of the leaf, one will add a unit to the number of leaves n' ; if the excess appears to be three-quarters of the thickness, one will add 1.5 and, finally, if the height is judged equal to twice the thickness of the leaf, one will add 2. A compound microscope, the eyeglass provided with spider lines and mounted on its lower side on a carriage furnished with a micrometer screw, allowing the object observed to move in the field of the instrument, can measure with great precision the height of the ogives of the wheels or of the excess of pinions. In default <rf this instrument, the method which we have just indicated is exact enough to be used. 283. The table which we give hereafter indicates the number of the division on the compass for the gearings most used in horologj-. Thus, on placing an So-tooth wheel at the divisii I Gearings, 195 TABLE FOR USING THE PROPORTIONAL COMPASS DBSIONATIOH HUMBKB OF TRKTH DIVISION OF THE COMPASS FOR SHAPE OF TEETH THE RADIUS OB DIAMETER — 1 SHAPE OF TEETH Elliptical Circular Elliptical Circular Wheel. . Pinion . . 180 12 183.542 13.66 1. 019676 1. 14 13-25 1. 104 Wheel. . Pinion . . 144 10 147.446 II.5 1.024 I.I5 11.05 1. 105 Wheel. . Pinion . . 96 12 99-747 13.66 1.03904 1. 14 13.25 1. 104 Wheel. . Pinion . . 80 ID 83.3853 11.5 1.0423 I.I5 11.05 1. 105 Wheel. . Pinion . . 64 8 67.1 9«45 1.048475 1. 18 905 I.I3 Wheel. . Pinion . . 90 12 93.614 13.66 1. 04016 I.I4 13.25 1. 104 Wheel. . Pinion . . 75 10 78.375 11.5 1.045 I.I5 11.05 1. 105 Wheel. . Pinion . . 60 8 63.0976 9-5 I.05II I.I8 9-05 1.13 Wheel. . Pinion . . 80 8 83.1247 9 5 1.039 I.I8 9.05 I- 13 Wheel. . Pinion . . 60 6 62.7839 7.4648 1.0464 1. 2441 7.05 1.175 Wheel. . Pinion . , 70 7 7 72.9637 8.397 7.972 1.0423 I. 1995 1. 139 8.05 7.7 1.15 real diameter 1.1 on pressing 2 leaves on one side and 1 on the other Wheel. . Pinion . . 48 6 50.77 7.4 I.0577 1.23 7.05 1. 175 Wheel. . Pinion . . 36 6 38.74 7.4 1.0762 1.23 7.05 1. 175 Wheel. . Pinion . . 30 6 32.72 7.4 1.0908 1.23 7.05 I.175 Wheel. . Pinion . . 36 12 38.55 14.02 Wheel. . Pinion . , 40 10 40.7 11.52 ig(5 /^isons in Horology. 83.38 of the compass, a loleaf pinion in which it should gear should correspond to the division 11.5 if the excess is of semi- elhptical shape, or at the division 11.05 'f ^'s form is semi -circular. For a pinion with the uneven number of 7 leaves, one will find two indications in the table, one giving the real diameter, the Other permitting the placing of the pinion with one leaf pressing against an arm of the compass, and the two leaves opposite against the other arm. This last measure comprises, therefore, in units of diametrical pitch, a total radius increased by the versed sine O B (Plate II). Alter what we have said, it will be easy to obtain in a graphical manner the figures corresponding to gearings not appearing : the table. 284. Verification of a Proportional Compass. The two divided scales should be perfectly straight and consequently in exact juxta- position when the instrument is closed ; this, one verifies by hold- ing the instrument to the height of the eyes and seeing if the twa scales are perfectly fitted against each other. The divisions should be regular and the zero point should be found in the center of the hinge. It is also easy to verify this condition with exactitude by taking off, with a pair of sharp-pointed dividers, a certain number of divisions, 10, for example : on moving, then, these dividers over the whole of the part divided, it is easy to assure oneself of the exact- ness of tliis condition. Finally, on placing one of the points of the dividers on the division 10, one should be able to place the other on the center of the hinge. This hinge should be made in such a manner that the arms can be spread without any jerk, that is to say, with even friction ; in no case could any play or shake whatever be allowed at this hinge. A compass being thus verified, it could be used with the aid of . the given table. There exist other systems of proportional compasses, most < which dispense with the use oi an accessory table. Let us remark, ' however, that the one which we have just described has its principle foimded on an exact and rational basis and that the table which it requires complicates its use \ery little, if at all 285. iKlerminatlon of the Distance Between the Centers of a Gcarinfr by Means of the Proporlional Compass and of a Depthing Tool- Having fastened the proportional compass in such a manner lof _ to« irt. ■ Gearings. 197 I I I I that the primitive radius of the wheel corresponds to the figure for its number of teeth, one measures, in this same opening, the diameter of one of the arms of a deptbing tool. Let d be the division corresponding to this last measure. One opens, then, the depthing too! until the two arms a and b, drawn in section (Fig. 76), correspond to the division ■^ + i. This opening then gives the distance between centers. Elxample ; Having regulated the opening in the proportional compass 50 that the total diameter of a 60- tooth wheel is fitted to the division 62.78 (seethe table), one measures the arm of a depthing tool and finds that its diameter corresponds to the division 8 ; we will thus have ^^+>- 4., the pinion having 6 leaves. The opening of the depthing tool should then be regulated in such a manner that the two arms a and b corres- pond to the division 41. 286. The Proportional Compass and Stem-winding Gearings. First — Gearing of the crown wheel in the ratchet wheel : The teeth of this gearing should be solid ; this is the reason why only one- twentieth of play is given them (223). For the same purpose the bottoms of the teeth are made with a rounded shape and the ogives of the teeth are shortened. These gearings are epicycloidal ; the profiles are formed by epicycloids described by a point of a gene- rating circumference smaller than half of the primitive circumfer- ences. As we have just said, one does not use the whole of the epicycloidal arc for the tooth ; it is sufficient that the contact be established three-fifths of the pitch before the line of centers, in order to be continued until tluee-fifths of the pitch beyond that line. 19^ Lettmu im J/aralogy. The "flank" of the tooth b no longer a stra^ht line, but a l^potycloid described by a point of the same generating drcle rolling on the interior of the primitive drcumlerence of the wheel Thus (Fig. 77) / fl is the useful epicydoidal arc, while a A is any curve whatever shortening the tooth ; in this manner, the height of the ogive is not determined ; / </ is a hypocydoidal arc gen- erally approaching, very nearly, a straight line. To determine the height of the shortened ogive in units of diametrical pitch, it is necessary to proceed graphically or by simply estimating it by the eye Generally, for this gearing, the height of the two ogives placed opposite to each other can be taken as 23^ diametrical pitch. If, then, n and n' are the numbers of teeth, the crown ^^-heel should be fitted to the division »' + aX and in the same manner the ratchet to the division « + aX- Gearings. 199 ^ » =rfr Let us note, however, that since the crown wheel always drives the ratchet, it is preferable to make the first proportionally greater than the second ; for example. Crown wheel, division . , . . n' -\- 2^ Ratchet wheel, division . . . w -j- 2]^ 287. Gearing of tlie Winding Pinion in the Crown Teetli of the Contrate Wheel. In these gearings the axes of the two mobiles form a right angle between them. Logically, such a gearing should be a conical gearing (311) ; in the practice of horology it is sufficient, however, to skillfully simulate it. One finds two general arrangements of this system. | In the first (Fig. 78). »blJ -" a ^ is the exterior diameter of the crown teeth in the contrate wheel ; this is, at the same time, its primitive diameter, for the ogive of the tooth is not to be added to the extremity of the radiufe, since the teeth are perpendicular to the plane of the wheel. The a:own wheel must, therefore, be fitted to the division w' of its teeth and the total diameter c d oi the pinion, perpendicular at i on n i to the division n + 2}^, as in the preceding case and also for the same reason. Therefore, Winding pinion, division n -\- 2}( Crown teeth of the contrate wheel, division . . n' The second arrangement is found in s Winding pinion, division . Crown wheel, division . . ne winding mechanism. It admits of a teeth range with the crown teeth outside of its primitive radius (Fig. 79) ; in this case there must be added to each of the two mobiles the height of the two ogives. One will then have 71' + 2 2O0 Lessons in Horology. 288. Gearlne of tbc Sliding Pinion and of tbe Small Setting Wbeel. Although one could not make use of the proportional compass for the study of the relative dimensions to be given to the mobiles of this gearing, and as this determination should be entirely a matter of calculation, we give here, however, the theory, which will not be found out of place. Suppose (Fig. 80) r to be the primitive radius of the small setting wheel, R its total radius and w the number of its teeth A. r' the radius of the sliding pinion abutting on the under side of the small winding wheel, R' its total radius and n' the number of its teeth. In the generality of cases, one can admit that consequently, ~ ' "' t' = R' ~ 0.2. If the penetration of the two mobiles is greater, the gearing- does not work well. Generally, it is desired in practice to deter- mine the number of teeth n' in the sliding pinion. The proportion ^ „ V ~ "^ gives us the value By an analogous reasoning to that of (286) one can place from whence one draws (0 Gearings. 2> Replacing r and / by their value in the equation (i). ecomes (=) I { Ji' ~~ o.a) ^ IJf' - 0.2) {n + a) Example : Let ^ ^: 2.4, jR' ^-- 2 and m =: 18, the formula (2) 289. Gearings of the Dial Wheels. The sma// and the larg^e setting wheels are placed in the proportional compass at the division corresponding to their number of teeth increased by 2. Therefore, Small setting wheel, division . . . . n +2 Large " " " ....«' -f 2. For the gearing of the cannon pinion and the minute wheel, s generally the same ; however, notice must always be taken oi I tiie form which the excess has, in the leaves of the cannon pinion. If the leaves are terminated by a semi-circular form, it would then be necessary, in this case, to place the pinion at the division corres- ponding to the number of leaves increased by i. Sometimes, also, the teeth of the minute wheel are formed in such a manner that one s obliged to add 2.5 or even j to their number, in order to obtain I the division of the compass at which this wheel should be placed. In order to verify at one time the series of relations between the wheels for setting the watch, the cannon pinion must be fitted to the division corresponding to its number of leaves increased by 3, and the other wheels, dial wheels and large and small setting I wheels, should likewise correspond to the divisions for their respec- tive number of teeth increased by 2. If there should be a fault, it is always better that the wheel which drives be a trifle large. Since one prefers, in this train, an easy and smooth transmission, and since the small setting wheel is the wheel which drives, one can allow it to be slightiy larger than I a strict proportion would give it The gearing of the minute wheel pinion and the hour wheel is sub- mitted to the same law with the same reserves (or the different forms of excess which are found in practice. Generally, however, one has Minute wheel pinion, division . . . . n +2 Hour wheel, division, »' -f 2, CI I I" r II t I t 1 I i I Lessons in Horology. Variotn Calculationi RelalfM b 290. The preceding table (283') gives a second factor for the various gearings which it includes, expressing the radius or total diameter of the mobile in function of the radius or primitive diameter equal to the unit The use of these factors is impor- tant in cases where it is desired to determine the total dimensions of the wheels of a gearing by means of a rapid calculation ; for example, in the construction of calibres. The solving of the fol- lowing problems will rapidly render us familiar with the use of this table : 291. Being given the primitive radius of a wheel to calculate its lolal radius R. To solve this question, the number of wheel teeth and of pinion leaves in which they should gear must be known. One seeks, therefore, in the second column of the table for the figures of the teeth range of the gearing, and the corres- ponding value indicated in the fifth column will give the factor by which the primitive radius of the wheel must be multiplied to obtain its total radius. One will, therefore, have ^ = r X factor of the table. If, for example, the primitive radius r of a barrel with 80 teeth is 10 mm. and if this wheel ought to gear in a lo-leaf pinion, the factor indicated by the table is r.0423 ; therefore, ^ = 10 X 10423 = 10.423 mm. If we had to calculate the total radius of a barrel with 96 teeth, whose primitive radius should likewise be 10 mm. and gear- ing in a i2-leaf pinion, we would have .ff = 10 X 1.03904 = 10.3904 mm. One sees that the total radius of the latter barrel is a little less than that of the lirsL 292. Being given the total radius of a wheel, to determine its primitive radius. This question is the inverse of the preceding one and consequendy is solved by dividing the total radius given by the tabulated factor. Therefore, ~ tabulated factor Numerically, one has, \l R ^^ 10.423, « ^ 80 and n' = 10, Gearings, 203 293. Being given the primitive radius of a pinion^ to calcu- late its total radius and^ reciprocally^ being given the total radius of a pinion^ to find its primitive radius. The same as for the wheel ; in the first case, one multiplies the primitive radius given by the factor of the table, and in the second, one divides the total radius by the factor. One has, therefore, Rf = r* Y. tabulated factor, or tabulated factor Just as for the proportional compass the table gives two factors for each pinion, one suitable for a semi-elliptical excess, the other for a semi-circular excess. Let, for example, r' = 1.25 and n* = 10, semi-circular form ; one will have ^j. R^ = 1.25 X 1.105 == 1.38025, ^_i:38o25 _ 1. 105 * If the excess was of semi-elliptical shape, one would have R' = 1.25 X 1.15 = 1.4375. or, for the inverse problem, I. 15 294. Being given the primitive radius of a wheels one seeks for the total radius of the pinion in which it gears (semi-elliptical form). One has the proportion r r" = n r* = r — n • 9 which gives the value and since ^/ = r^ X tabulated factor one obtains, on replacing r* by its value, R' -^r — X tabulated factor. n Thus, as example, r = 5.38. « = 75, «' = 10 one will have R' = 5.38 X ^ X 1. 115 = 0.8249. 304 Lessons in Horology. 295. Being given the primHive radtus of a phdon^ one desires the total radius of the wheel We have n and '-^ ^ ^^rxtab. &*. = ,- -^XtaKfac Let r* = 0.86, n = 80, x' =: 10. One will write 80 R = 0.86 X — X 1.0423 = 7.17. 296. Being given the total radius of the wheels to find the total radius of the pinion. We have tab. fac of the wheel and ^ «'. n on replacing, it becomes n tab. fac of the wheel Afterward R' ^ T^ tab. fac of the pinion ; consequently, „, „ nf tab. fac of the pinion n tab. fac of the wheel ' Let, for example, R = 10.2, « = 80, «^ = 10, Tabulated factors i r». . -43 ( Pmion = 1. 15 one will have R' = 10.2 X -^ X -^^- = 1.4067. 80 1.0423 ' 29 T. Being given the total radius of the pinion^ to find the total radius of the wheel. In an analogous manner to the preced- ing case, one will have ^= ^' and from whence And mtu'<? one hrts rtt last tab. fac. of the pinion n r ^ r" n' R' n tab. fac, of the pinion nf R =z r tab. fac of the wheel, n tab. fac of the wheel R R' n' tab. fac of the pinion and since one has Gearings, 205 Let R^ = 1.4067, n = 80, n' = 10, one will have r> 1.4067 X 80 X 1.0423 /t = ■■ r-z = 10.2. 10 X 1. 15 298. Being given the dista7ice of the centers and the numbers of teeth in a gearings to determine the total diameters of the wheel and pinion. We know the formulas (185) r = D — I — 7-, and r* ■= D \ ,> n-\- nf n -\- nf R = r tab. fac. of the wheel, R^ = r^ tab. fac. of the pinion, ft R = D i > tab. fac. of the wheel, n -X- fr nf R^ =^ D 1 7 tab. fac. of the pinion. n ~x- n D = 5.2, n = 70, n^ = 7, Tabulated factors I ^'^^^^ ^^'' ^^^ ^^^^^ labulatea lactors | j j^^^ f^r the pinion one will have , ^ _ 5.2 X 70 X^i.0423 X 2 _ ^ g^^^ , ^. = 5j_X7_X 1. 1995 X 2 _ 77 ^ 299. Being given the total radius of the wheel and the num- bers of teeth of the gearing, one desires to find the distance between the centers of the two mobiles. The formula r ^ D j — 7 gives __ n -^ n^ and smce R Let .,, t_ tab. fac. wheel one will have _ ^ n + n^ tab. fac. wheel ' n Suppose o „ / „ ^^ R == 4.927, n = 70, n^ = 7, Tabulated factors of the wheel, = 1.0423. One places _ 4.927 X 77 _ , , 1.0423 X 70 300. Being given the total radius R of a pinion and the num- bers n and n' of the teeth of the gearing y to determine the distance of the centers D, I I In sn analogous write the formula Suppose lessens in Hifrology. to the preceding i t99S X 7 301. Being given the diameter P of a walck plate, Ike num- iers of teeth n of the barrel and n' of the center pinion, one desires b> find : ist, the primitive radii r and r' of the wheel and of the pinion .- 3d, the distance of the centers D of the two mobiles ; jd, the total radii R and R'. The diameter of the barrel should be as large as possible. This question is generally one of the first which presents itseSf in connection with the establishing of a new watch calibre. In order to be able to fit the plate of a watch in its case, a "recess" is generally made on the exterior oi this plate, which can be valued at one- sixtieth part of the total diameter P. There remains, therefore, only ^ available. The useful radius b, consequently, ' 60 120 The extremity of the teeth range in the barrel can coincide with the extremity of this radius, the teeth finding the necessary play in the hollowed-out part in the center of the case. The radius -j-Vff ^ should be equal to the sum of the following lengths : the primitive radius r" of the center pinion, the primitive ndius of the barrel and its total radius (Fig. 81). One would, therefore, have one will also have Gearings, 207 from whence one finds r = _5?. 120 n -|- I -f tab. fac. Knowing r, it is easy to determine r', since r'= r n the distance of the centers D will afterward be determined by the sum of the two primitive radii. Fig. 81 Knowing r and r', one will calculate the total radii R and R! by the operation : R = r tab. fac. of the wheel. R* z= f' tab. fac. of the pinion. Suppose, as an example, /* = 43 mm. (19 lines), « = 80 and n' = 10. Tabulated factor of the wheel = 1.0423. Tabulated factor of the pinion, = 1.15. One will have : 59 r = 120 X43 i + I + 1.0423 2o8 Lessons in Horology, or also and It follows that Then and finally Since one has ^^ 59 X 43 ^ 59X43 lao (i + I + 1.0423) 260.086 r = 9-7545. D = 9.7545 + 1.2193 = 10.9738, R = 9-7545 X 1.0423 = 10.17. ^ = 1. 2193 X 1. 15 = 1.4- ^-43 = 21.1416. one should have likewise, or very nearly, r^ -]- r + /^ = 21.1416. The addition gives 1. 2193 9.7545 10.17 21.1438. With this approximation the result is satiskctory. 302. Being given the total radius R of the pinion in which the rack of a repeater gears^ the number n* of teeth cu:cording to which the pinion is divided^ and the total radius R of the rocky one desires to find the number according to which the sector of this last xcheel must be cut. Let us admit the ogive of the teeth range of the two mobiles equal to twice the diametrical pitch (277). We will have the primitive radius of the pinion by the formula and that of the wheel r^ = r = R' n" w' + 2 R n Sii\oo one has n T ^ r* If' - ~» one can also write r IT n' ir' -f 2 R' n" {n + 2) nr A^ n /? i» (i/+ 2) ft ' 1 *f>n<^ Vn<^wit X\\aX th« diviftton of this pinion br ih^ siet of teeth is not complete, for the «>Ni»iM( thAt thin moMl<» oul^r 4>x<khU«« « f^Actlon of « turn. The pitch of the gearing left fUI tllotUtiitM (ho Mrix>»t of U)<» woT^m^pnU n simplifying, R («' + 2) = /" (» + 2). I One finally obtains \ Suppose, for example, R ^ 9' 961 R' = would have : « = 9 9^ X ^5 _ _ , _ ft, ,„,u Remark.— One could obtain directly the above formula by a iropordon analogous to that of 289, 303. The following problem does not find its solution in the theory of gearings only, but also in that of trains and of the motive force. It recalls to our mind, in an excellent manner, the P Studies that we have gone over, so we do not hesitate to close this ■eries of problems by joining together some of the various ques- tions which we have treated in this chapter and in those which have preceded it. 304. Problem. A horologist has constructed a stem-winding watch the diameter of whose plate is 50 millimeters (22 lines). ^H Upon winding the watch, he notices that the power necessary to ^^koperate the winding works, that is to say, to overcome the force ^^■lOpposed by the spring to the movement, is too great. He decides ^^B'then to manufacture a new watch, like the first, but in which the ^^^winding can be more easily eflected. He should, therefore, modify ^^Vthe value of the two factors which enter into the expression of the mechanical work ; the force on the one hand and the iime employed for the winding on the other ; in other words, the spm:e traversed by the point of application of the active force (37). »The first watch has its barrel furnished with a stop work of 4 turns, and runs for 32 hours ; the second should run for the same number of hours. It is evident that if we introduce into the second watch a barrel furnished with a stop work, allowing it to make 5 rotations during 32 hours, we wilf have diminished the average tension of the force and augmented at the same time the ^H,duration of the winding. ^ Let us seek, therefore, lor the nature and the value c change that must be made in the second construction, in order to \ arrive at the end desired. SOS. In the first place, the relation between the numbers of teeth ] in the barrel and of leaves in the center pinion must be changed. In the first watch, this relation was \^, which gave a length erf J running -- X 4 ^ 3* hours. In the second watch, one should also have -^- X 5 = 3= hours. On choosing a pinion of lo leaves, one will have K = 6.4 X lo = 64 teeth, or a pinion oi 15 leaves, n = 6.4 X 15 = 96 teeth. It is not an absolute necessity, in general, to conserve above relation in a very strict manner. Thus, if one wished a pinion with 12 leaves, one would have n = 6.4 X !2 = 76.8 teelh. This fractional number not being practical, let u example, « = 78 We would then have '- X 5 = 325 hours. The watch would run, with the above number, half an hour longer than was desired. Since, for a watch of 50 mm. diameter, a barrel with 96 teeth does not give too weak a teeth range, one can accept for this gearing 96 teeth for the barrel, 15 leaves for the center pinion. 306. The primitive radii of the two mobiles must now be cal- culated and the distance between their centers. Let us commence by seeking for these values in the first watch, in order to compare the results, Gearings. 211 The formula which we have determined (301) gives us + I + 1.039 96 and making the calculation ^ ^ 11.36. Then . «^ r v^ 12 r' = r — - = T1.36 X -7- = 1.42, and n 96 D = 11.36 + 1.42 = 12.78. The total radii of the two mobiles were : R = 11.36 X 1.039 = 11.803, R =^ 1.42 X 1. 14 = 1. 619. For the second watch, the calculations are naturally analogous, only we have here the case of a gearing whose teeth range is not found in the table. One can admit, in this case, by analogy the tabulated factor of ^he wheel equal to 1.04 by slightly forcing the figure of the factor "^ the preceding gearing, since the height of the ogive should '*^ crease with a larger pinion (15 leaves instead of 12). For the pinion, we will concede an excess with semi-circular ^*Xape perfectly admissible for this number of leaves. We will thus have 59 r = 120 X 50 ^-nd making the calculation ^ + 1 + 1.04 r = 11.17; and , 15 -Therefore, ^ D = 11.17 + 1.745 ■= 12.915 ADne will have the total radii R = II. 17 X 1.04 = II. 617 ig^ = r^4- ^ \f = r^ (i -\- -^^ = 1.891. 15 X 5 \ 37.5/ ^ One sees that the diameter of the barrel has diminished and that of the pinion and the distance between the centers has increased. 30t* Let us now seek for the exterior and interior radii of the barrel drum. These dimensions ought to be as large as possible. The ex- terior radius of the drum in the first watch was 1 1 mm. , therefore 312 Lessom in Horology. radiu That of the second i 0.36 less than the primitive 1 could, therefore, be 11.17 — 0,36 = lo-Si mm. The Ulterior radius of the drum was, in the first case, 11 — 0.77 = 10.33; consequenUy, it could be, in the second case, 10.3I — 0.77 = 10.04 or 10 mm. 308. Let us now calculate the dimensions of the hub, and thc'^ dimensions as well as iheyurce of the spring. In the first watch, when the spring is pressed against the side of the drum, it occupies the third of the interior radius of the barrel, another third remains empty and the third third is occupied ■ by the hub. This spring makes, in this position, 15 turns and 4.5,'! placed loosely on a table : the number of turns of development is 6.5J The dimensions of the blade are the following : Thickness ^0.21 Height ^3.9 Length = 780 n According to the formula (97) ^ E h e' i the moment of the force of this spring if F = 557-64- When the spring is coiled up, that is completely wound, one has >i = 15 + 6 — 4.5 = 16.5; consequently, p ^ ,^^ y, ^^^ _ g,^ g,. When the spring is down, one has n' = IZ.5 and F' = 12.5 X 557 = 6942 gr- in order that the center wheel may receive in the second watdlJ the same force at the beginning and at the end of the tension of t spring, it is necessary that, the watch being wound, one should h ^1' = -|- X 9190 = 7352 gr- and if the watch is run down ■'^i' = -7- X 6942 = S553 er-. Gearings, 213 since for the same number of teeth in the barrels the relation of the leaves and pinions is ^^ . ^ "~ T' In order that this difference between the moments of extreme force may exist with 5 turns of the stop work, one must have the proportion ^6^ ^ «/ + 5 12.5 «/ ' from whence one finds , , «/ = 15625, and afterwards «i = 20.625 — «. 309. Let us further calculate the thickness of the spring. Let e be this thickness for the spring in the first watch and ef that of the spring in the second. For the first case we have the moment „ E h e* 2 11 n ^= 12 L and for the second 4 ^ £ A e^* awjn on remarking that the length L increases in inverse relation to the diminution of the thickness e. One could, therefore, place 4 £ h e' 2wn £ Ae^' 2ir ^ n "5 12^ " 12 L. S ' from whence, on simplifying, one obtains 5 ^a ^^»=-t and therefore, from whence 5 4 25 ^* = 16 e\ \ 25 \ 5 For e = 0.21, one has e^ = 0.21 X 0894427 = 0.10826. 310. Since we have the proportion L e" '214 Lessons in Horology. one could iinall]r calculate the new length of the spring on placing^ L' = 871 mm. Since the second barrel is a little smaller than the first, this spring will fill it a little more than one-third ; but as it is also thinner, one can diminish the hub proportionately to the relative thickness of the spring (m). Thus, the first hub having a radius equal to one-third of the interior radius of the barrel, this one would be -y- ^= 3.66 . . mm. This radius being 17.777 times greater than the thickness of the spring, the second hub could l>e 17.777 X .18826 = 3.34.. .mm. 31l> We know, moreover, the means of increasing the ease oi the winding by increasing the number of teeth in the ratchet and of the contrate teeth range in the crown wheel ; since we have already treated this question (169), we will not go back to it here and we will thus admit that the problem proposed is solved. Conical G««rin^. 312. In the gearings that we have just considered, the two axes are parallel to each other and we know that the movement of the system can be compared to that of two cylviders mutually conduct- ing each other by simple contact We have designated gearings oi this sort under the name of cylindrical gearings. 313. If, in place of being parallel, the two axes are concurrent, one can imagine thai the movement of one produces the movement of the other by the contact of two cones concentric to each of the two axes (Fig. 82). This system takes, therefore, the name of conical gearing. The two axes can form any angle whatever with each other ; wc will treat particularly the special case where this angle is a right angle, almost the only case in horology. Suppose (Fig. 82) O x and O y, two perpendicular axes iiround which turn the two cones COB and A O C\ let us admit thnt their movement is produced without slipping. Gearings, 215 As for the cylindrical gearings, one can prove th'it the rela- tion of the angles traversed by the two cones is inversely propor- tional to that of the diameters C B and A C The diameters can be measured in any manner whatever, pro- vided that their circumferences be tangent. Thus, in place ol C B and A C one can just as well take C B* and A' C\ since these straight lines form the sides of similar triangles. .x' Fiff. 83 » and «' being the angles traversed in the same time by the two cones, one has, therefore, the proportion A C ^ A ' C __ a' C B ~ C B' ~ \' and since, n and n! being the numbers of teeth one will also have a' a n It' A C n C B n' 314. The pitch p of the gearing varies with the distance O C; for such a point of contact C, it is AC B C n nf 2l6 lessons in Horology. 315. Form of the Teetll. As in the cylindrical gearings, the* transmission of the movement cannot be effected practically by the simple contact of the two primitive cones ; one is generally obliged to supply these cones with flanges, that is to say, with teeth, which malce them move as if they were driven by their simple adhesion. The contact, consequently, is not always found on the line O C; if we consider the contact at the point C, the displacement of this contact should take place on the surface of a sphere whose center is at O and passing through the points A C B. The form of the j teeth must be traced on this sphere. Thus, M O and O N (Fig. 83) being the two axes of rotation 1 which meet in a point O, let us take this point as center of a | sphere ; it will contain the two upright cones, as we have just said, having their common summit at the center O, and will cut them I ■long two circumferences, of their bases, tangent at the point A I belonging to the generatrix of contact of the two cones. These circumferences drive each other, exactly as would the 1 primitive circumferences of a cylindrical gearing situated ir ae plane. The sphere playing the role of the plane consideredj X. 217 in the first case, all the p ropert i es aheady determined are here reproduced. Thus, one can determine the cnnre described by a point of one of these primitive drcnmferences by making one of the cones roll on the other which remains immovaUe ; the curve thus a^es- dered is the spherical epicycUnd B C. On account of the similarity of the methods employed in this case with those that we have previously described, we will not enter into all the details of these constructions. 116. Let us examine, howe\'er, the case cA JUmk gearings. The flank being a diametrical f^ane of the primitive cone, the drivii^ tooth will be a conical surface whose form must be determined. Suppose S O and S (J the axes of the two primitive cones which should turn while touching along an ec%e S M ^not repre- sented in the Fig. 84 ^ Suppose Mm NzdA Mn^ N" the circmn- ferences of Ihe cones by Ihe dieir proceetiiag fe-ocn the rntersectioo of drawn pterpetwixcnlarly throcgfi rfie On die radios J/ <7 or due cnrcie a dicassS/txfsat CT zodi ihcuughics M h8 Lessons in Horology. 1 its plai ; this perpendicular i C one erects a perpendicular o meet the axis at a point S. II one considers this point 2 as the common summit of twaa cones having for bases the two circles O and O", and if one make9l the second cone S O" roll on the first S <3, a point of the < ierence (?" will describe a curve m m" , a spherical epicyclaHA situated on the sphere on which is moved the circle O" sphere having its center at S. If one made a cone which had its summit at the point ^ p through this epicycloid, this cone will be the exterior surface of a diametrical plane of the cone ^ O and should consequently be taken for the surface of the cone ,^ O. This result appears evident from the similarity in the construction of cylindrical gearings, there- fore, we will add no other proof to the application of this development by analogy. 3ir. Besides the cycloidal iorm, one ploys also the evolvent ol circle for the teeth of coni- cal gearings. 318. Construction fA Conical Gearins's. By the preceding, all the lines which enter into conical gearings being defined, it is only necessary to apply the principles of descrip- tive geometry to deduce \. y irom them the outlines nec- -^ .— '^ essary for its construction. Fly. 8s But it is useless to enter into extended details wilJI' regards to this, since in practice a more simple method been adopted and one sufficiently exact. In order to represent these forms in a more convenient manner on paper, one substitutes for the spherical surface the plane surface projected zX F C G (Fig. 85) perpendicular to the line of tangency of the primitive cones O C. On this surface I Gearings. 219 one represents the developed surfaces of the two cones A G C and B FC{¥\g. 86). Ill tlie development of the cone O D, the circumference prxj- jecled in A D C will become an incomplete circle A C A with radius .^ C of the same length as this circumference. Likewise, the cone O E developed will give an arc of circle of same length as the circumference projected in C E B. It is on these circumferences that one lays oft the lengths cor- responding to the pitch of the gearing determined on the circum- ferences with radii A D and C EkA the bases of the two cones. One draws afterward the form of the teeth of the two mobiles, as one does it on the primitive circum- ferences in the cylindrical gearings. These forins are obviously equal to those that one would ob- tain on the spherical surfaces them- selves, since, for the small dimensions of a tooth, the surface of the plane and that of the sphere are almost the same. For the purpose of being able to compare the form of the teeth of a conical gearing with that of the drawing, one terminates these wheels by a portion of the cones C G A and C5 /^(Fig. 85). 319. In horology, however, one cannot do this, either on account of the lack of room or because the wheel carries at the same time another teeth range, as the crown wheel of stem-winding gearings, for example. The exterior surfaces o( the two wheels are then straight planes, perpendicular to the axes. Admit, for this case, that the exterior planes of the two mobiles meet in C(Fig. 87), and let us seek for the form of the teeth cut by the plane C B. It is, in fact, on this plane that we see the form of the teeth range and that we can determine its dimensions. Let us first draw the two primitive cones COB and A O C. the latter being represented only by its half DOC; draw the mdicular F Gxq Cand a parallel F' G' to F G, finally the 220 Lessons in Horology. primitive circumferences with radii F' C and G' C* and determine the form of the teeth according to the method known. In order to obtain a horizontal projection of the tooth of the wheel (winding pinion, for example) whose center we can place at A : % • % % \ : % 1 di" i— •—•?"•-* •-•■^ "• 1 .^ y *^ ' «Ci . v^ » v» Jm y y -' ^*^ . fen !'^.'»*. -' -V^ . ' « \\ • • • '•'.♦ V *'-y^ ' \ ■( . ■■■'i .v:..\ ,••* ()", let lis note that the point C is projected at Cand C", we will lmvi% therefore, the circumference of the primitive base of the cone wllli mdiuH E C •-- O" C"; let us lay of! half the thickness of the Ippih C il* on each side of C" on the primitive drcle. Gearings, 221 In order to determine now the total radius (7' /", correspond- ing to the point /', let us project this point /' on the plane F G at /; draw the radii O /prolonged to M\ the point of intersection M of this radius with the plane B C gives the total radius E M that one can project on O* /"; from the center O* describe after- wards the total circumference of the wheel. One proceeds in an analogous manner to determine the bottom of the teeth on the plane B C hy projecting the point K' on F G, drawing the radius O K cutting the plane ^ Cat iVand projecting this point at K" ; one will have thus the radius O K'* of the cir- cumference passing through the base of the teeth. In order to determine finally any point /*" of the form of the teeth, project the point P' to P on F (7, draw the radius O P U and project the points P U on (9" /". From the center (7' one causes to pass through the points obtained arcs of a circle and lays off the half thickness ^S*' P' on the circle passing through ^S*"; and 322 Lesiont m Horology. draws, afterwards, the radius O" T". The fx^t P" at which this radius just cuts the circle projected from U, is the point of the tooth. Let us stitl seek for the form of the teeth which appears on the interior plane L Q parallel to the plane B C M {Fig. 88). On the side elevation, the point Q represents the point of the teeth. Project this point on the front elevation, at Q", and describe from O" as center the circumference which passes through this point and which gives us the point of the teeth. To obtain the point T" of the base of ihe teeth, draw the radius O N, cutting the plane L Q 3.1 T and project this point on the front elevation, we will thus obtain the point T" through wliich one passes the circum- | ference of the base of the teeth. I In order to further obtain any points whatever, for example, those which are found on the primitive cone, one draws the radius O C, cutting the plane I. Q aX. V, projects this point on the front elevation, at V" and describes the circumference from the center O". One afterwards draws the radius O" R ; the points determined by 1 the intersection of this radius with the circumference passing J through V", are points of the curve of the teeth. One can thus I determine as many points as one desires and represent in thiaf manner the complete form of the tooth. Let us remark that, compared with the form determined for the teeth on the plane F G (Fig. 87), the form obtained on the front elevation having 0" as center, is elongated. One should, therefore, take account, in practice, of the elongated shape of the I teeth in these wheels compared with those of corresponding planefl gearings. The drawing of the front elevation of the wheel is made i exactly the same manner. Defedi which PrMent Themielves in theie GcBrin^i. 320. When, in a gearing, the normal to the point of conta does not pass through the point of tangency of the primitive cii^J cumfercncos, the transmission of the force is irr^ular. The faults of construction which most often produce effect, are : First — A relative disproportion between the total diameters d the two wheels. Second — A gearing too close or too slack. Third — A bad te*;th range. Gearings. 233 According to the case, one will then find in the gearing a ' ' butting " or a " drop. ' ' 321. The butting, also called binding, is the irregular contact of two teeth before the line oi centers. If, for example, a is the point of tangency of two primitive circumferences O 0| and ff (Fig. 89) and c the /'"' \ ""•■■._ point of contact of a tooth and a leaf, one will find on drawing the normal to this point that in place of pass- ing through a it will cut the line of centers at a point a' situated between a and O . There will result a diminu- tion of force transmitted at this instant for the two fol- lowing reasons : First — In place of a force F' ^ -^ §^i ^"^ *''" have only F' ^= F -^^, . as much different from the first as the point a' is found nearer to the center C. Second— Increase of the re-entering friction. The causes which can produce this defect a (i) Two slack a gearing; (2) A pinion proportionally too large ; (3) A bad teeth range. Fig. 89* shows the case of too large a pinion ; the pitch of the gearing is longer than that of the wheel. The tooth B has ised to conduct the leaf and the tooth A enters too soon into contact with the succeeding one. As we have said, the moment of the force transmitted is, therefore, diminished. Fig. 90 shows the case of too slack a gearing. In place of entering into contact with the straight flank of the leaf, the tooth Oi e generally : le tfae dtfecti thai tfaef d be uf teai-eUlpticBi to ^ Bppreclflbie for piuli conducts, first, the excess, the normal cuts the line of centers be- the center O oi the pinion, the case oi a bad teeth range ; the tooth, too short, for example, has its contact with the leaf, as in the preced- " ing case : the normal passes between the t I point a and the center O and one has a dimi- nution of the force transmitted. When the above defects are not too much accentuated, it is possible to remedy them, in order to ob- tain a passable g'ear- ing ; but, at least in the first case, it is im- possible to arrive at absolute perfection, If the pinion is Fig. 90 slighdy too large, one can touch up the wheel in such a manner as to free the teeth range at the base a b (Fig. 92) and make it less pointed, after the manner of the English teeth range. If the gearing is too slack, one increases the diameter of the wheel by careful forging. If the teeth range is defective, one can try to rectify it by means of a suitable ordinary cutter, or, still better, with an Ingold cutler. 322. II the first contact of the tooth with the leaf com- mences after the line of centers, it may happen that at a certain moment of the movement the angular speed of the wheel becomes proportionately greater than that of the pinion which it conducts. This defect is the drop ; it is produced by (i) Too close a gearing. (2) A pinion proportionally too small. (3) A bad teeth range. Gearings, 225 Fig. 93 shows too small a pinion ; the pitch of the gearing of the wheel is greater than that of the pinion. When the tooth B should cease the contact on the generating circle, the tooth A is still found removed from the leaf that it should conduct. The tooth B will slip along the flank of the leaf and at this instant the normal to the point of contact will not pass through the point of contact of the primitive circumferences, but will cross the line of centers at a point nearer the center of the wheel ! One will, therefore, \ have, in this case, an ^ increase of the force transmitted. For a uniform movement of the pinion, the wheel will take an accele- rated movement ; this is, technically speak- ing, a * * drop. * ' Fig. 94 represents too deep a gearing, the tooth B conducts its leaf farther than the generating circumfer- ence ; there is, there- fore, produced a slip- ping of the point of the tooth against the flank of the leaf, the accelerated movement which the wheel takes terminates by a drop of the tooth which follows on the leaf which it will conduct. The direction of the normal at the point of contact shows that one has, in this case, also an increase of the force transmitted. Fig. 95 represents the case of a bad teeth range of the wheel. The teeth, which are too long, drive the pinion leaves farther than they should geometrically ; one can thus recognize the drop which will be produced. A gearing presenting the above defects can Fig. 93 be corrected by diminishing the height of the Fig. 91 226 Lessons in Horology. ogive in such a manner that the teeth drive the leaves a less d tance or, otherwise, by forming the teeth in such a manner as give them a greater breadth on the primitive circumference. 323. Oi of a wheel ! just done, that the gearii^ pinion produces a butting, that, the other hand, too deep a gear^ j ing produces a drop, one i j that it is best to make a de^ gearing when the pinion is t large. Reciprocally, a gearing who9( pinion is too small should 1 relatively shallow. 324. A defect which oni encounters often enough in gear- ings is that which is by pinions whose lee long enough, that is to say, pinions which are not cut t ! enough. If the teeth of the wheel are correct, one finds very 0* often the point of the tooth i Fig. i);i contact with the bottom of the leaves (core of the pinion), one cannot change the pinion, which is the only means to obtain a perfect gearing, the ogives of the teeth must then be shortened^ either by cutting off the points or by modifying the shape. One understands that in these cases absolute perfection exists no longof, especially if the number of pinion leaves is small ; since then thu contact should commence before the line of centers. 325. One encounters very often, also, pinions of ordinar] quality in which the flanks of the leaves are not directed toward the center, but are diverted more or less from it Such piniooE should be rejected as much as possible if one wishes to preserW in the gearing the quality of a flank gearing ; if not, the tooth a the wheel would have to be formed by means of a curve describel as we have indicated (215). In a gearing, detective either on account of the shape or direo tion of the pinion leaves or the wheel teeth, if one modifies one a the two profiles it might happen that one arrives at a coned Gearings. gearing fullillmg all the con- ditions of a uniiorm trans- i mission of the force, even \ when the essential character- ^ btics of the flank or epicy- doidal gearing no longer exbt. In this case, the entire theory of the determination of the . forms of contact is there in order to make us understand that one has luckily been able to find a combination of forms fulfilling the condition estab- lished, that the normal to the successive points of contact passes constantly through the point of tangency oi the primi- tive circumferences. We know that this condition suffices for the gearing to be perfect, whatever may be the shape of the profiles established. Paiiive R«iuluic«t In Gearing. 326. General Ideas. We have already indicated that the pas- fflve resistances are forces which naturally present themselves in all machines in motion(46). 2J These resistances are of diverse natures : some pro- ceed from the bodies them- selves, from their weight, their form, their dimen- sions, and also from the relativeness of the move- ments which animate them. Such are friction, and its congenerics, inertia and shocks. Others arise, more properly, from the medium in which these bodies are moved, such as, especially, 0' 338 I^ssffns in Horology. the resistance of the air. Among tliese, the principal cause c absorption of work which is to be considered here is the friction., I of which we will first take up the general study before applying J the laws to the particular case of the gearings. 327. When a body is moved by slipping on another body, there is produced a resistance which is opposed to the movement. This resistance is due to the action of the two surfaces in contact, when the movement already communicated to the body allows the inertia to be excluded. This resisting force is friclion ; it appears to proceed from the reciprocal action of the molecules of the two bodies. The inequalities of surface more or less evident in thesej bodies penetrate each other reciprocally, fit into each other witha much greater intensity in proportion as the two bodies are more* closely pressed together. Moreover, when one of these bodies ^a\ displaced the resistance produced by this "binding" is hirtherl increased by the driving back ol the molecules situated in front of 1 the moving body, 328. Besides this cause of resistance, there exists a second one^B due to the adhesion of the two surfaces. The effect produced 1 this second cause can be made very apparent by placing on eacJi:l other two planes of the same kind ; if the surfaces are very care-J fully planed and perfectly polished, as, for example, those of two J mirrors, the adhesion may become so great that the separation rfj the two bodies becomes very difficult.* Friction depends, therefore, on the two causes mentioned ; but^l the last is very often neglected if the two surfaces are directly in I contact, that is to say, if there is no coating or lubricating sub- \ stance, such as oil, between these surfaces. Numerous experi-j mcnts have, in fact, proven that this resistance may be neglected! when the extent of the surfaces in contact is not very great. 329. But when one interposes a greasy substance betweeni the two bodies, it is no longer possible to neglect this last cause, I which, in certain cases, may diminish the friction properly ■ speaking and, in others, increase it. We will treat, further « of this question and will hmit ourselves for the moment to th«i| study ol "dry friction." • Thl« pbonnmeiion arlaea from Ihe mora or lew mmpleta eipulaloii of [he air l)Mire««.f Gearings. 229 330. The Two Kinds of Friction. If the same part of the surface of one of tfie rubbing bodies always remains in contact with the other ■ body, there is shding, and the friction takes the name of "sliding friction." If, on the contrary, the surfaces in contact change at each instant, there is rolling and the friction takes the name of "rolling friction." An example of the first case is the friction which is established during the movement of a sleigh along a road ; and of the second case, that which is pro- duced when a wheel rolls on a plane. ^We will occupy ourselves especially with the sliding friction, only kind which we will encounter in horology. 331. The slipping may be linear, that is to say, be efiected aiong a plane or any surface H-hatever when one of the bodies is continually displaced with relation to the other ; or it may be circular, if one of these bodies turns on itself without going forward, for example, a trunnion in its bearings. The friction of the teeth of a gearing is produced by a linear slipping ; that of the pivots of these same wheels in the interior of the holes in which they turn is produced by a circular slipping. 332. Laws of Friction. It has been discovered by very care- ful experiments that the resistance due to friction is subject to three principal laws which can guide in the applications and which are sufficiently exact within the limits between which they are con- sidered in machines. ^^m Vivsl— The friction is proportional to lite normal pressure ; ^Hftiat is to say, the resistance is always the same fraction of the pres- ^^^Bre which applies one body on another, which is easily understood, ^^Bbce the actions of the molecules should arise by reason of this ^^Bressure. ^^V Second — The friction is independent of the surfaces in contact; ^^This is to say, when this extent increases without the pressure changing, the total resistance remains the same, although the pres- sure on each element of surface is found to be diminished in inverse relation to the extent of these surfaces. Since, for given sub- stances, the friction is a constant fraction of the pressure, it fol- lows that a heavy body drawn on a plane gives rise always to the same resistance, whatever may be the extent of the surface of contact. Third — 7^ friction is independent of the speed of the move - ^^^ment; which is to say, that the same amount of work is necessary »30 Lessons in Horology. in overcoming the iriction o( a body traversing a certain distance, no matter what may be the speed which animates the body. By the aid of these three fundamental laws and of the values determined, experimentally, in order to establish the relation of the friction to the pressure according to the nature of the surfaces in contact, one may value in each case the work absorbed by friction. 333. Experimental Determination of the Force of FricliOD. Let U9 mippose that a body with weight P be acted upon by a force F which makes it slide with a uniform movement on a surface A B (Fig, 96). Ont' knows that when a body is moved uniformly, TOwwrownrawrawTOw; lig. I the algebraic sum of the forces which act on this body is equal to zero. The force F should, therefore, be equal, and in contrary senHC to, the force of friction : it will be, therefore, the measure of the greatness of this resistance. One of the laws which we have cited, showing that the friction is proportional to the pressure, it follows that if the weight P is ^_ doubled, the friction is doubled at the same time and consequendy ^H its equivulent F. ^H The relation —^ is, therefore, constant for the same substances ^H in contact : this is called the "coefficient of hiction," that is gen- ^H erklly represented in the calculations by the letter/. Thus one has t When this coefficient is known, as well as the pressure P extended nonnally to the surfaces in contact, one can determine the Irictioa F by multiplying the pressure P by the coefficient / Tlwiirfiwe, F = f p. 3M. Let us note that the co^Bcient -A friction does Dot always keep tbe sune wKw tor diffcnnt surttces <A the same knid. for tke Gearings. 231 r a body is and the more it is polished, the less is the friction. Its value is also modilied by interposing a greasy substance, oil, for example, between the surfaces in contact. The object of this operation is principally to avoid the grating and the heating of the frictioning bodies. One knows, in fact, that without this precaution there are detached from the surfaces small fragments which groove them deeper and deeper ; the friction speedily increases and the heating which results from it can even go so far as to make the bodies red hot and to set them on fire if they are combustible. One finds that friction of steel on steel produces by the grating a reddish dust, which is, probably, oxide of iron ; the dry friction of steel on brass enables us to prove that a certain quantity of brass is deposited on the surface of the steel ; the heating should, in this case, be considerable. Horologists know the grooves, often very deep, which the lack of oil on the pivots produces, when these turn a long time, dry in their holes (the fact is especially noticeable on the pivots of the center wheel) ; they are familiar also with the deep lines worn in the leaves of tempered steel pinions, caused by the teeth of the wheels made of gold alloys, which, for this reason, are almost entirely abandoned in these days. One sees by these examples that a high speed of the mobiles is not necessary to produce the grating, which is on the whole entirely in conformity with the third law of friction, 335. The following table is intended to give an idea of the mean value of the coefficient / in the most general conditions. It is best, in each particular case, to choose this value properly, according to the probable conditions of the action of the parts in motion. .„„.„„.,.„ =BI.ATI0J./nirTnKFBIO Metal on metal Metal on precious stones .... Wood on wood Bricks and stones on the same . . Leather tiands on metallic pulleys . CIS to 0.17 0.IS 0.33 0.6s 0.30 to 0.40 933 Lessons in Horology. In large machines whose frictioning parts are carefully greased the coefficient of friction diminishes to a value of / = 0.08. 336. Work of Friction. The mechanical work of a force being the product of this force by the path traversed by its point of application, when the path and the force have the same direction, one will have, if E is the path traversed, W. }^ ^f P E. If the two bodies are movers, it will be necessary to consider the two forces of friction which have / P for common value and which act on each body in the inverse direction of its movement with regard to the other, each of the forces producing work. Suppose that the movements of the two bodies A and B (Fig. 97) are effected in the direction of the arrows (i), ^and E' being the respec- tive paths traversed in this direction, which will be that of the relative movement of the ^ mpm body A, if E i greater than E'. If we consido! the movement the body A the friction will produce a work / P. E., which 1 be negative, since the direction_/'/'is the inverse of that of £■; t is, therefore, a resisting work. On the contrary, the work of j5 o A will produce on this first body a positive work y /". E', will take away from the resisting work /P. E,, so that finally th« resisting work produced by the friction will be W. F --=^ f P E — / r. E' = / P {E — E'), E — E' being positive. If E— E' becomes negative, the equa-^ tion no longer holds, and the direction of the friction must 1 changed ; in place oi having f P (£ — E'), one will have Let us here note that the work developed by the friction ( one of the two bodies is positive ; with regard to this body, there-^ fore, friction plays the part of motive force. This property ii employed industrially in the transmission of movement by cylin- J ders, cones or friction plates. Gearings. 233 33?. Anfle of Friction. Suppose a body resting on an inclined plane A C (Fig. 98) ; let us admit that we have regfuialed the inclination oi this plane in such a manner that the body may be on the point of moving, or, what amounts to the same thing, that it is moved with a uniform motion, the length of this plane. In this case the force of the friction is equal and in a contrary direction to the force which acts to make it descend. ^H The weight p of the body acts along a vertical line e a, pass- ^mg through its center of gravity ; making c a ^ p, drawing the line c b parallel to the plane and c d perpendicular to this direction one will be able to form the parallelogram of the focus by drawing the lines a d parallel to ^ 1: and a b parallel to c d. The length c b will then represent the value of the force F tending to make the body descend along the plane, and the length, c d, the normal I pressure P. We will, therefore, have FIk. S8 % : and B C A give, moreover, the ^ = — =/. The similar triangles & proportion ^ ^ Vi ^ One can thus see that the coeilicient of friction is equal to the quotient of the height B C oi the inclined plane divided by the length of the base A B. 338. Designating by +, the angle, CAB, that the incHned plane forms with the horizon when the movement takes place, the two components of the weight p can also be represented by p. sin + -. f- following the direction of the inclined plane and by »Jt Lessons in Horology. perpendicularly to this plane. One has, therefore, F / sin + . . , One discovers, on varying arbitrarily the extent of the in contact and the weights of the bodies, that the angle of inclii tion does not vary for the same substances in contact This angle is called the angle of friction and the numerical value of the relation of the friction to the pressure, equal to tang. ^, is the coefficient of the friction. For hard and polished metals and the stones used in horology, this angle has a value varying front 7" to 8" 30'. Example of Application. — On a plate of tempered and polished steel we place a ruby, a lever pallet, for example. We elevate little by litde one of the extremities of the plate until the ruby commences to slide with a uniform movement. The height B C Xo which it was necessary to elevate the steel plate being I. -5. 4 mm. and the length of the base A C Sg mm., the coefficient of friclion of the ruby on the steel should be The angle of friction will be, in this case, tang. 4 ^ 0.15 and + =; 8° 32', value that we will adopt in our calculations. C&lculKtion of the Friction In Ge&ringi. 339. Knowing the normal pressure between the teeth of al gearing, one has the value of the sliding friction (333), so that if onfr I knows the length of the space traversed by the friction of one tooth^ on another, one would have the work absorbed by this friction (336). J Before entering into the details of this calculation, let us reca the kinetic question of the transmission of the movement. We have found that by means of gearings, the movement of one wheel is uniformly transmitted to another ; this geometrical demon- stration is independent of the material of which the wheels are formed, of the nature of the friction, etc. This property holds good whatever may be the friction in play and the greatness of the efforts which are shown. The passive resistances have, therefore, no eflect on the trans- mission of the movement, properly speaking ; they only increase J I s. 235 the work to be expended in onier to prodoce the movement of the motive whed. One thus understands that the work of fnctioa may be generally expressed as function ci the resisling useful work to which it is added. 340. Let us adopt the k^owii^ notation : A, the whed which oontrob the movement ; r, its primitive radius ; n, its number ci teeth ; A\ the whed contn^led ; r^, its primitive radius ; n', its number of teeth a =^ t c = t d (F%. 99) the pitdi of the gearing, and Q the resis- tance opposed to the movement ol A acting tangendy to the primi- tive circumference. 0. 0' Generally, in gearings, there are several teeth of the whed which work at the same time ; but, in order to facilitate the calcu- lation, we will suppose that there is only one and that it controk from / to ^, that is to say, a space equal to the pitch. During this passage a, the work absorbed by the normal pressure between the teeth is equal to thb pressure multiplied by the length of the curv^e traversed by the point of contact / in its passage from /to /', a length which does not perceptibly differ from a^. But, since the * Let U8 note that the normal pressure has not generallf a constant ralue ; it would bare it onlj in the case of gearings by inTolrent of circle, if one neglected the firiction. For the others, it is Tariable and it is the mean Talue of this quantity that must be made to enter into the expression of the work of the friction. »36 Lftions I Hffrologjr. I \ point of contact is very slightly removed from the arete', one can sappOK, without committing an appreciate error, that the pressure Ui equal to the force Q acting tangently to / t', and the work absorbed is. therefore, „ g From the hypothesis that the normal pressure between the teeth ia constantly equal to Q (mean value of this pressure), i follows that the friction is equal to /0 As to the work absorbed by this friction, one remarks that while the point of contact / passes to /', the space traversed by this point on the tooth which controls is equal to i' c and that which it has traversed on the controlled wheel is t' tf. from whence it follows that there has been sliding on a length equal to a difference which one can suppose equal to a straight line joiaii c to ir'. The work absorbed by the friction is, therefore, Q/X c C. Dropping the perpendiculars c e and c" e' to the line of centers; O being almost parallel to c d , one can suppose but, one has tn admit - and / f' = — c t ^ c' I = a. which comes nearer the truth as the pitch becomes smaller wil relation to the radius, one has, therefore, consequently the work absorbed by the friction is The work which the wheel A should transmit to the wheel A? for the distance traversed a is, therefore, If- - Gfl + G«4" irV + 7^) "' ■A nhord l> the dimk iin]KirtloiuU batman the dimnuUr lud lu prajHUgn Gearings. 237 If P is the motive force which acts tangently on the driving wheel, the motive work for the distance traversed ais P a and one has from whence 34L Taking up again the formula ( i ) in which Q a represents the useful work '' Wu /' this can still be put under the form Since one has, moreover, T n and n' being the number of teeth, and since n a ■= 2^ r and «' a = 2 ir r', one obtains ^ ^ ^ ^ r = and r' = 2 IT 2 IT On replacing these values in the formula (2) and simplifying, it becomes Wn.= Wu ^ Wu ./» (-1- 4- -^)» (3) or from whence one draws 342. On examining these two last equations, we can see that one diminishes the friction by increasing the number of teeth. Thus, for a wheel of 64 teeth gearing in a pinion of 8 leaves, one would have, on admitting Wm = i andy= 15, Wu = + 0.15. 3.i4i6(-^4--|-) 1.066 while for a wheel of 96 teeth gearing in a pinion of 12 leaves, one would have only 1.044 1+0.15. 3Ui6(-^ + ^) »38 Lessons in Horology. One thus discovers the practical rule that th& number ol must be increased as much as possible in order to diminish the of friction, to have less wear and a smoother mouon. 343. For interior gearings, the formulas (3) and {4) becomi ^ W, + Wu 1 +/ (s) (6) One sees that in these gearings, with the same number of teedl the friction is less than in the exterior gearings. 344. The friction in the rack can also be deduced from the preceding formulas, on remarking that the radius of one of the primitive circumferences becoming infinite the general expression dt' the friction, for « ^ 00, becomes »'«=»'-+ »■» /"4-. (7) and ' + /«- (8) 345. For conical gearings, on preserving the same relationi as in the preceding cases, one would arrive at the following results, a. being the angle formed between the axes of the twd wheels : . - «. + »4 ^?^ ^ + ^ + '-^^ (9l m = - Wm (10) the These gearings are smoother than cylindrical gearings of the same number of teeth. 34^. Friction Before and After the Line of Centers. formulas which we have established, the influence of the friction is the same before as after the line of centers ; this result does not agree with those of e.vperience, which show, on the contrarj-, that the, friction before the line of centers is more hurtful than that which exerted after the passage of this line. However, since we have! supposed the pitch as being very small, our results can be coi sidered, in this case, as sufficiently exact. h is certain that it would no longer be the same if the contai commenced at a relatively great distance from the line of centers. Gearings. 239 347. Let us examine, for example, the case of a wheel with 60 teeth gearing in a pinion of 6 leaves, since we know that in horology this gearing is one of those in which the contact of the tooth and the leaf should commence the most in advance of the line of centers ; and let us examine successively the four following cases : First — ^The wheel drives the pinion after the line of centers ; Second — ^The pinion drives the wheel before the line of centers ; Third — The wheel drives the pinion before the line of centers ; Fourth — The pinion drives the wheel after the line of centers. 348. The Wheel Drives the Pinion After the Line of Centers. Suppose O and Cf the centers of two mobiles (Fig. 100), P the 240 Lessons in Horology. moment of the force with relation to the axis of the wheel which drives, and Q the moment of the force with relation to the axis of the wheel which is driven. The wheel is in equilibrium under the action of the force P, whose moment is P, of the normal force N' =^ A' whose moment \s, ^ N. O b and of the force of frictiony^A^ directed perpendicularly to the normal force, and whose moment is — f N. O d. One has, therefore, P~N0b-fN0d=0. On the other hand, the wheel O is in equilibrium under the action of the force Q, whose moment is Q and of the normal reac- tion N' = A^ whose moment is — N' . & b'. The moment of the force of friction is null, since its lever arm is equal to zero. One has, therefore, On dividing the first of these equations by the second and simplifying, one obtains ~Q (y b' But, on designating by p the angle b t = V t a formed the normal and the line of centers, one will also have by P r. sir P+/('-+ .^)cosP (0 Q r'. sin P and, on dividing by sm p, g- - -5- (. + ^'^ "■"«!:?> or also ^-^ = ' +/ ( I + -J) "^otang p. Let equal to us remark that if, ero, we will have in this equation, we make P r' the friction Q " •- an analogous formula to that which we haveestablished(i96,equa. 17). If, in the above formula (i), one places Q r ^ i, one will obtain ior P r" a value superior to unity. The angle a (Fig. 100) which the leaf is diverted from the line of centers, is the complement of p ; one can, therefore, also write the equation (1) : Gearings, 241 Numerical Calculation. — Let Qr= i,/=o. 15, ^ = -^. • = 42» 15' Af^ one has " °" Log: 0.165 == 0.2174839 — I Log: tang a = 9.9584454 0.1759293 — I Number . . . = 0.14994 We will therefore have the relation P r" -Q-^ = 1. 14994. On subtracting the friction, and admitting the moment P = I gr. , we would have, in this case, P_ ^ ^ from whence Q 6 * e = i.-^=o.igr. On introducing the force of friction, one will have P ^ 1.14994 X 60 from whence Q ^ Q = = 0.08696 gr. 11.4994 ^^ ^ 349. The Pinion Drives the Wheel Before the Une of Centers. In this case the moment Q becomes the moving power and the value Q r will become superior to P r*. The formula (i) then becomes ^ Qr _ I I — / ^ I + — tang a J remarking that the sign of the friction is changed. Numerical Calculation. — The same data as in the pre- ceding case, except that we take here P r = i. We have / A -f -^^ tang, a = 0.14994, then ' -^ (^ + ■?■) ^"^- • = ^•^5006, Or I , 242 Lessons in Horology, On subtracting the friction, one would have, if i^ = i gr., -9- = A, from whence ^ 60 ' P= Q^-^ 10 gr. On introducing the force of friction, one will have = 1. 1764 from whence ^ 60 * P = ^-z- = 8.5 gr. 0.11764 ** ** 150. The Wheel Drives the Pinion Before the Line of Centers. We have in this case (Fig. loi), reasoning the same as in the preceding cases, P — N. O b = O, Q — N'. a 1/ ^ f N\ a d = O. from whence P_ _ r. sin P . or, again, 8 ~ K sin. p -/ (r + r^ ) cos p ' Pr^ T Qr 1-/(1 + ^) cotang p But the angle b O (y, complement of p, is equal to a — . n since the angles traversed in the same time by the two mobiles of a gearing are inversely proportional to the numbers of teeth (176). One will, therefore, have, Pr^ _ I Qr ~ f ( y ^\ ^ / ^\ ^^^ -/(.+-^)ung(.-^) Numerical Calculation. — Let « = 17° 44' 13", j» = 60, »'- 6. One has i. -- = ai X 17^ 44' 15" = i^ 46' 25.3" /("i "^ I ) ^^ 0.15 X II = I 65 k\^. 1.05 = 0.2174839 lo^. rang, (i^ i?) - S.400S94S 0,70837^7 — 2 Number ^= 0.051095 Gearings, 243 I — / ( I + i?) tang. ( tt i) = I — 0.051095 = 0.948905 and = 1.05384 = Pr" 0.948905 ^^^ Q ^ If the moment of force /* = i gr., we have, without the friction, nf 6 ^*- % % ■ ^' \ I \ I \ \ \ \ \ \ (h >V ! Fig. 101 On introducing the friction, one has Q = from whence 1.05384 ■ n 1.05384 Q = 0.09489 gr. 10 244 Lessons in Horology. 35L The Pinion Drives the Wheel After the line of Centers. The moment Q becomes the moving power and the formula (3) becomes n r / n \ / n'\ on changing the sign of the friction. Numerical Calculation. — One has, from the preceding^ calculation, -p-p ■= 1 + 0.051095 = 1.05109s. Without the friction, we will have from whence ^ * P = O ^-, andif 6= i gr., ^ '^ ^ With the friction, one will have P^^ . ' ^ TO 6 " 1. 051095 1. 051095 and /• = 9-514 gr- 352. Recapitulatlan of the Preceding Calculations. The moment of the motive force acting on the wheel being equal to I gramme, the moment of the resisting force with relation to the axis of the pinion should be at the instant of the first contact before the line of centers, Q = 0.09489 gr.; and at the instant of the last contact after the line of centers, Q = 0.08696 gr. When the pinion drives the wheel, we have found at the instant of the first contact before the line of centers, /•-8,5 gr., and at the instant of the last contact after the line of centers, P = 9-514 gr- One sees that, in the most usual case, when the wheel drives- the pinion, the force absorbed by the friction before the line (A centers difTers very litde from that which is absorbed after the e of this line, which confirms what we have admitted (261),* r lo b« bHb lo oomparp fbeni \n sn i IrlresUe pinion, the mnniFDl of Ibe fa n is inftrior to that -'-■-'■ — >■ •— dleuUtcd for eiplsios itbj, «b«o Gearings. 245 One sees also that the smaller the driving wheel becomes with relation to the one which is driven, the more also increases the dif- ference of the resistance before and after the line of centers ;^e numbers of teeth of the two mobiles should, therefore, be ipcfeased [ifis much as possible. Colcutatiooi of the rriction of Pivott. 353. When the watch is placed in a horizontal position, the E diSerent mobiles of the train rest on the flat ' ' shoulders ' ' of their TUV lower pivots ; in the vertical position, these same mobiles rest on the cylindrical surfaces of the two pivots. The force of friction b proportionate to the pressure which the surfaces in contact undergo. In the horizontal positions, the pressure proceeds from the weight of the mobile and from the lateral force which presses the pivots against the sides of the holes ; in the vertical positions, these same forces are in action, but on account of the position of the gearing on the axes, these pressures will generally be different on each pivot. 354. Work Absorbed by Friction on the Plane Surface of the Shoulder of a Pivot. Suppose (Fig. 102) r" the exterior radius of the shoulder and r" the radius of the pivot. Repre- senting also the mean radius --^ — by 8 and the width of the shoulder by /, then / = r" — r. 346 Lessons in Horology. One will have, consequently, ^' = 8 ^ i and r' = S — -. The surface of the circular crown with radius r" — r' being r (r"' -^n. one will have The surface of the circle with radius r' will be P being the pressure exerted on the crown and admitting that thia i pressure varies proportionately to the extent of the surface ; that I which is exerted on a circle with radius f' should be I v^v(=' . the circle with radius r" ') and that which would take plact would be p / The work absorbed by the friction of the crown is equal to J the work absorbed by the friction which would be produced on the 1 total surface of the circle with radius ^' ^ S + j- diminished by f that which would take place on the surface of the circle with j radius r" ^ s — -j' ; it is, therefore,* or ^, • Knr ■ pnuuie P. the force deieloped by rricUon it f P; Ih* " pradaot of//>bi the dlal»ce irnrerwd, Ttus dlsUoce li not the >Bnie for nil Ihc polnli vtM DbUJo lU mean •slue, divide 'thF olrcle wUb radlns r iota t number n of euui] Kclon 1 ■utBelaaUjr ■nmll ao IhU eacb of them dao be regatded u a Iriingle. The reBu^Iaol of tb« I ([rii*llT. "T. «t K n' 'h* radius. The force of (HcHoo being for one of them / - , ' ,^.2„^ r-ft. J»r. W- i fFw r. Gearings. (s.i)_(.-|)., !•/'■(,' + r + — *7-5 — ) it becomes, therefore, k*'- ••/' (i « ^ s-j) -"""(' + As) 3SS. In the horizontal position, the pressure P arises from the weight of the mobile ; this pressure is always much inferior to the lateral pressure with which the pivots are pressed against the sides of the holes. Thus, in the preceding equation, one can neglect the term ., and one has simply " I W = f P. 3 » 8 ==/ Pw (r" -f r*). B 356. In the vertical position of the watch, the pressure P on ■■ the shoulder of the pivot is null ; one can, therefore, also admit that the friction is null. The work of the friction of the cylindrical surface of the pivots against the sides of the holes is expressed by I W = f P. ill r'. " r* being the radius of the pivot. The formula includes the work absorbed by the two pivots, since P is the total pressure and since the friction depends only on this pressure and not on the extent of the surfaces in contp.ct. 3ST. Determination of the Lateral Pressure Received by the Pivots of the Mobiles in a Train. l,et us examine, for example, the third wheel of a watch. This mobile receives on one side an action on the part of the center wheel gearing in its pinion and, on the other, a resistance arising from the pinion of the fourth wheel in which the third wheel gears. These two efforts show themselves by a pressure on the axis, and the two pivots are pressed against the sides of the holes ; for each of these, the load which they receive can be represented in magnitude and direction by the 24^* Lessons in Horology. resultant of the partial forces that the axis of the wheel receives at each of its extremities. This pressure depends on the relation of the distance of the pivots from the point of application of the forces in play to the length of the axis. 358. Let us imagine the point of contact of the teeth and the leaves on the line of centers and let us represent by P the force that the leaf of the pinion receives on the part of the whed tooth of the center wheel ; since, simultaneously with this, one wheel tooth of the third wheel is pressed against a pinion leaf of the fourth wheel, the force P gives birth to the reaction /". The direction of the forces /"and P' is perpendicular to the line of centers. Let us call r and r" the primitive radii of the wheel and pinioo, we will have for the state of equilibrium P r' = P' r aa6 P' = P ^. The equilibrium will not be disturbed if one will apply to the point diametrically opposed to that of the contact of the tooth and leaf a force P^ equal, and in a contrary direction, to P and, like- wise, at the point opposite to the point of application of P', i force P\ equal and in a contrary direction to this one. The resultant of the forces P and P^ is 2 /* ; it should be; applied at the axis of the pinion, paraSlelly to the components (Fig. 103). The top pivot of the third wheel will receive on the part of the resultant 2 Pa force fi and the lower pivot a force q in such a w« that one would have ^ /» = a -I- g One should have p a — q b. a representing the distance of the shoulder of the top pivot froi the middle of the thickness of the center wheel, and b the distanra from this last point to the shoulder of the lower pivot. To determine the values / and q, we extract from the first formula q = 1 p - p. from whence, on substituting this value in the second, J Gearings. iK4i ^*Vv* /W^/-* ^ **m:> O B A *fXBsL a3& 1:1 «. "HTe to! isi^ a r^ ^f^^/^ O // A O U ff ,taABA = /. sks O.^ = 5 ^; /^, >/^ ^^j4^'Hfy, // ifft^i // \fy iiht haU of their Taloe, 00c obuios , , -r * r tf -r * cos«» COS a. I'iff IIm- luw^f |ilvo(, onr would obtain in an analogous manner W^i ll|Min I'HfUiilnlnM lh« Imj^. 103, one will notice that the pUHMMh' nt ll»i» M'nliM whrrl tcrth is greater than the resistance w\s\\\\ ll\»« li'i^vrftot thr» \\\\\\\ pinion oppose. Consequently, foi \ C tarings. the top pivot of the third wheel, the force O C ^ p will be greater than the force O B ^^ p' . For the lower pivot, the pressure oi the center wheel diminishes, the force O C :^ q becomes weaker, and O B" ^^ q" stronger. There necessarily results a different direction of the resultants R and ^1, which the construction of the parallelogram of the forces shows. The direction of these resultants is important and enables us to explain the reason why one encounters, in making repairs, pivot holes enlarged by wear in a direction often very different from that where it would seem that this wear should logically be produced. This fact is noticed in clocks or pocket watches whose holes are not jeiveled. 360. Let us note, moreover, that the problem we have just dealt with is based on the case of flank gearings when the con- tact takes place on the line of centers ; the normal forces are then perpendicular to the radii. But when the contact between the tooth and the leaf ia displaced or when the gearing is of I another sort, the normal forces take other directions and the J value of the resultants, as also their directions, can undergo a I slight change. 36L The equations (i) and (a) show that, all other things being equal, the pressures become greatest for an angle a ^ 180°, for which cos a — — i ; the sign of the last term placed under the radical becomes then positive. From this standpoint it would, consequently, not be desirable to construct a train, all the mobiles of which would be in a straight line. 1362. We establish also that the more the value of r increases, the more the pressure diminishes ; this is one of the reasons why it is good to increase as much as possible the diameter of the center and third wheels, inertia having not yet appreciable influence on these mobiles, the mo t I 363. Finally, it may not be useless to observe that the relation ,- cannot be replaced by the relation of the numbers of teeth, since the mobiles with radii r and r" do not gear together, but are mounted on the same axis. 364. Numerical Example. — Suppose P= 77-5 gr. fl = 0.8. b = 4.06. d = 3.2. e = 3.a. r' = 0.87. r = 5.49. a = 93°. 252 Lessons in Horology, We will have for the top pivot P 77-5 ^ 0.87 r— .- = ~^- = 15.95 • = = 0.16, a -^ d 4.86 ^ ^^^ r 5.49 and d* = 4.06* = 16.40. ( — ^ ) "^ (0.16 X 2.2)* = 0.12391 2 a d — - cos a = 2 X 2.2 X 4-o6 X 0.16 X — 0.08715 = — 0.25. and ^ = ^5.95 1/ 16.48 + 0.1239 + 025 ^ = 15.95 1^16.8539 = 65.47. For the lower pivot, one will have successively J^i = 15-95 -Jo.S' + 0.16 X 3-2" — 2 X 0.8 X 0.32 X 0.16 X cos 95^ ^"^ -/?! = 15.95 1/ 0.64 + 0.26 + 0.07 ^x = 15.95 1/0^97 = 15.71. 365. Let us now determine the value of the work of friction of the third wheel's pivots during one oscillation of the balance. We have the formula (356). IV. F = f Pr, p. in which P represents the pressure, r^ the pivot's radius, p the angle traversed during one oscillation. Let us first seek this latter angle. The fourth wheel makes one rotation in 60 seconds or in 300 oscillations. If the third wheel has 75 teeth and the fourth pinion 10 leaves, this pinion turns 7. 5 times faster than the third wheel ; the third wheel, therefore, makes one turn in 300 X 7-5 oscillations = 2250 oscillations. During one oscillation it will traverse, therefore, an angle p : P = -^— = 0° 9' 36^^ ; 2250 ^ ^ this angle expressed in length of arc with radius equal to unity is Qj. 0° 9^ 36-^^ •= 0.00279 0.0028 in round numbers. The diameter of the pivots being 0.26, we will have for the top pivot IV. P= 0.15 X 65.47 X 0.13 X 0.0028 = 0.0035 gr. mm.. and for the lower pivot IV. P = 0.15 X 15.71 X 0.13 X 0.0028 = 0.00085 gr. nim. Gearings. 253 The total work absorbed by the friction will, consequently, be W. F= o.exijs + 0.00085 = O.OQW5 gr. mm. The work of the motive force applied to this wheel being I 0.2 1 gr, mm. during the duration of one oscillation of the balance, I one can prove that the work absorbed by the friction of the pivots \ represents about the fiftieth part of it, Influence of the Oil. 366. We have said at the beginning of the study of friction that the introduction of a greasy substance between the frictioning surfaces of two bodies, compelled to slide on each other, is neces- sary in all cases where a heating and consequently grinding and ir are to be feared. When greasy substances are interposed between two surfaces, I these are no longer in immediate contact, the molecules of grease liform little spheres which roll between the two bodies. In most I cases, especially in large mechanisms, the friction will be reduced I by this fact. In horology, especially in pocket watches, the inverse phe- ftnomenon can present itself. The oil which is used introduces a new ■.resistance, an adhesion, or, otherwise expressed, a "sticking." ■ This new resistance is added to the friction and it can happen that Ithe coefficient of the sura of the two resistances may be greater than Bthe coefficient of dry friction. With regard to the weak forces in I action on the last mobiles of the train, on those of the escapement l.and on the balance, this last resistance cannot be neglected. Unfor- ftunately, it is very difficult to express this force in figures, because ■it depends on the nature of the lubricant, on its degree of fluidity liand on its unchangeableness. The friction which is exerted through the agency of a lubricant F depends on the speed of the bodies in contact, and on the extent of ' their surfaces. It depends also on the nature of the movement ; thus, it is different on an annular balance when the latter is animated with a continuous circular movement and when it is animated with an oscillatory movement (circular reciprocating). One under- stands, in this latter case, that a certain quantity of oil participates j' in the movement of the pivots and that this oil would have a ten- I dency to continue in the direction of the movement, although the I pivots turn already in the opposite direction. I Lessons in Horology. In all the experiments relative to the friction of lubricated bodies, care must be taken to assure oneself that the lubricants are neither altered nor expelled. One can take as a general principle that the best lubricant is that which is the most fluid, that is to say, it is better, when one can, to replace grease by oil, oil by water, water by air, which is equivalent to suppressing all lubricants. This supposes that the speed of the mobiles may be sufticiendy great not to expell the lubricant experimented with. But a considerable speed is necessary for the pieces to retain a fluid lubricant like water and with still much more reason for them to leave between themselves a sutfident cushion of air. Experiments have been made with the astonishing result of showing the friction almost suppressed between two pieces nibbing together without any lubricant and at an enormous speed.* This almost entire disappearance of friction is due to the inter- position of a cushion of air, a perfectly elastic matter, between the surfaces in contact. In horology, in all cases where the use of a lubricant is neces- sary, one must, therefore, take into account the speed of the mobiles and the pressure which they have to support. Thus, the wheels of the stem-winding mechanism should always be greased by means of a semi-fluid lubricant, f The motive spring, as well as the pivots of the arbor around which the barrel turns, should be lubricated with a thicker oil than that which one employs for the train and the escapement. The principal qualities of the refined oil which is used in horology should be its unchangeableness by the atmosphere and by the various temperatures which the watch must stand, its perfect flnidity and the absence of acids in its composition. The solution of this question so important to the preserving, for the longest pos- sible time, of the precision in the running of chronometrical instru- ments, lies within the domain of organic chemistry. blghiy ft ■Dd reBDcd t Gearings. 355 Application of the Theory of Gearings. Functioni of (he Heut in Chronofiaph*. 36?. Chronographs are horary instruments intended to measure very small intervals oi time. For this purpose these watches are furnished with a special hand fastened at the center of the dial and traversing a division generally exterior to the minute circle. The shortest interval of time measured by the hand of the chronograph is equal to the duration of one oscillation of the halance ; thus, when the balance of the watch beats 18000 oscillations per hour, the chronograph indicates the duration of an observation to about one- fifth of a second.* These mechanisms are of many different constructions ; their movement is controlled by the train of the watch, causing, by this fact, a slight additional burden to the motive power. Before the observation, the chronograph hand is fastened and remains on the division zero. On pressing an exterior push-piece, this hand is immediately put into motion ; at the end of the observation, a second pressure stops the hand and, finally, after reading, a third pressure brings it back suddenly to the division zero, where it remains held in place until the moment of a new use of its function. The invention of the chronograph goes as far back as the year 1862 and is due to Adolphe Nicole, originally from the valley of Lake Joux but established in business in London. 368. It does not belong to the plan of this study to give a description of the mechanism composing this instrument ; it will suffice for us to show that the action which returns the hand to zero on the division, is effected by the fall of a jumper on a heart-shaped eccentric fastened on the axis of the wheel which carries the chronograph hand. We will especially occupy our- selves here with the determination of the form to be given this eccentric. The condition which the heart should fulfill is to present, at every point of its outline, a sufficient inclination to the lever (or jumper) which works it to assure the slipping of the lever as far as I the origin of the curve. • C. W. Scbmldt, x Svpdlih englaeer, IIiIok Id Paris, hu coostrualed chronogriplu Meue >ad gtop, are loteaded eipeclallT to meaniTe tlie ipeed oi prniectllei. 256 lessons in Harology. On imagining the axis on which the heart is fixed animated with a condnuous circular movement and the extremity of the lever constantly pressed against the exterior border of the curve, the problem becomes, to find the form capable of changing, uniformly, a continuous circular movement into a reciprocating ■ circular movement. 369. Lei us first examine the simpler case of the transforma- tion of a continuous circular movement into reciprocating rectilinear movement, by means of a heart-shaped eccentric. Since the reciprocating movement should be uniform, the point B of the line A B (Fig. 104) should successively occupy the J equidistant positions B, i, 2, 3, 4, -i, the lengths B i, 12, 2j, ^, . . . . being supposed equal fracUonal parts of the total j course B A. If, from the point O as center, one described cir- cumferences passing through the points B, i, z, 3, ^ A, and if one divides the circumference whose radius \s O A into I the same number of divisions into which the line A B has ( been divided, the intersections of the circumierences, with the radii passing through the points of division, will indicate succes- sively the points through which should be described the envelope curve of the point B. By construction, the uniform movement kMk I ! Horology. of this curve around the center O will communicate a unifomt rectihnear movement to the point B, alternately from AtoB from B to A, tlie lower part of the curve naturally being symmet- rical with the upper part. One recognizes thus that the form obtained is that of a spiral of Archimedes, elongated, whose equa- tion is % = aw^ C: the radius vector S less a constant quantity, C, is always propor- tional to the angle described, w. 370. In chronographs, the reciprocating movement of the point B is rot rectilinear ; it is circular and its movement ift executed around a center C (Fig. 105). Let us, therefore, now determine the shape of the heart suitable for the above new condition, and let us make the pointed end of a lever (7 B traverse an arc A B with a uniform movement, while the axis O turns the arc B D C^ ^ r^. Let' us divide the arcs A B and B D C into the same number of equal parts, and let us describe from the center O concentric circumferences passing through the division points of the arc A B draw afterward the radii passing through the division points of the 3XK. B D C. In order to determine, now, the points m, n, o. A' and' m' , n', o', of the curve of the heart, let us consider, as in the preceding case, the intersections of the circumferences and of the radii, but let us lay off here in addition the lengths of the arcs included on each circumference between the initial radius B E and the arc descrit>ed by the point B of the lever, before the points of intersection considered. One would thus obtain the curve B ?n n o A' o' n' m\ which would fulfill the con- ditions desired. In fact, supposing, for example, the point o of the heart arrived at the point j of the arc B A. the radius K will be superposed on the radius B E, the axis of the heart has completed three-quarters of its course and in the same time the point B of the lever will have been lifted up three- quarters of the distance B A. 371. Let us decide to determine by calculation the value of the radius vector of the heart corresponding to the point of the lever for any position whatever of the axis of the eccen- tric (Fig- 106). Gearings, 259 3 Fig. 106 3 Lessons in Horology. Suppose ; , ihe distance between the center of rotation of the heart and that of the arm (^R should also be the distance from the point of contact to the center of the arm) ; , the variable radius vector from the center of the heart to any point whatever of the exterior curve ; , the radius vector of the heart corresponding to the position repose, when the hand of the chronograph is at zero ; , the greatest radius vector of the heart ; , the angle formed by the radii vector r^ and r ; nt of centers O (7 and the radius R contact with the radius vector r contact with the radius r contact with the radius r \ B. the angle formed by the li of the arm when the latter of the heart ; e„, this angle when the arm is e', this angle when the arm is The radius r is, therefore, the chord of radius R and corresponding to angle e ; one r = 2 V?. sin i 0. The angle a, formed by the radius vector r^ and the radius considered r, is different from the angle which the heart should turn starling from the position of repose to the instant when the radius r coincides with the point of the lever. Designating thufl last angle by -y, we have in effect T = » according as the axis turns to the left or to the right, p being t angle formed by the fixed direction of the radius r^ and that whicj the radius r lakes at the moment of contact with the lever. Let us^ P'ace p ^ J (fl _ fl^), this angle being inscribed in the circumference with radius j and e — flj angle at the center embracing the same arc. We w3t J have, consequendy, ^^ ^ + (e — e„). Let us now establish the relation which connects the angU^J e — e„ and -y, angles which should be in a determined relation, 1 virtue of the mechanical principle stating that the transmission e the force is uniform when the angles traversed in the same time b two mobiles which drive each other remain constantly ii relation. But when the lever traverses the total angle 8' — iJ Gearings, 261 the heart executes a demi-turn, therefore, an angle equal to v ; one will, therefore, have ^ ^ from whence ^^»(e_e,) Consequently, one can write e — e and ^' "~ ^^ e^ -K a = '»r ^,---V^i(e^0^) and also w Oo '^ »o from whence e a 4- w e^zre"" =^ i ^o = » j73:t" "^ * *» One will thus have e The equation of the two branches of the curve of the heart expressed in polar co-ordinates will, therefore, be r = 2 i?. sin i (—^ -h eo^ 372. Numerical Calculation. — Let us admit r^ = 2 R. sin J 6^ = 4. r^= 2 R. sin J 6-' ^ 24. 7? = 140. We will have . . ^ 4 , . , a, 24 sin 4 80 = -~ and sin 4 8^= -~, Which gives ^280 ^280 •o = 1° 38' ^Z^' consequently, •' = 9" 5c/ a^'' ; 8^ — 80 = 8° 11^ 49.4^^ Expressed in seconds of the arc, the angles 8^, 8^ — 8^ and v give ®o = 5893.6 seconds 6^ — 60 = 29509.4... ** IT = 684000 . . . ** Let us first calculate this equation under the form r = 2 and suppose a = 30®. ^ sin i (^,qr xfe^^^ej + •-^ Lessons in Horology. i(e'-«<,) = ■ log: 648000" log '4754-7 662754.7 The angle a - log : 108000 + i < y - fl. ) t«+ i (»'-fl.)) =5-8213528 — log : (8' - fl) = 4.4699589 1. 35 1 3939 : 30° ^ 108000" = 50334238 4808.72 ■ therefore, 3.6820299 log: 4808.72". i.72 -i- 5893-6 ^ 10702.32"; from whence log log : 2 .^ sin I B r = 7.36327 for > S.4139741 2.4471580 0.8611321 Similar calculations will give successively r = 10.5254 for a = 60 r = 13.786:; ■' a = 90 r = 17045' " a = 120 r = 20.3017 ■■ B = 15c r = 23.5623 ■' o == i8c For the other branch of the r ^ 2 Jf s\a\ s the formula would be C-^ra^^'^'O' and identical calculations to the preceding would give the follow results : r = 7-4153 'or B r = 10.8294 r ^ 14.240 ;- = 17.66 >- = 21.06 r = 24.4648 For this last calculation, the radius r no longer belongs to the closed curve, but rather to the prolongation of this curve. Let us remark that the greatest radius vector should be equal to 24. Gearings, 263 If one would like to know, also, the value of the angle p, cor- responding to the above data, one would have 373. In order to obtain a greater stability for the chronograph hand, one prefers, sometimes, to make the heart like the adjoining form (Fig. 107) ; one suppresses, in doing this, a part of its curve, but, by way of compensation, the part B A B' oi the arm is terminated by curves fulfilling the conditions desired. Fli;. 107 Note.— In the original French, paragraphs 374 to 377, indn* sive, appeared at the end of Vol. I as an appendix, but in trans- lating were moved to their proper places in the text and the pan^praph numbers, together with the figure numbers under cuts, were accordingly changed as follows : Paragraph No. 374 now appears as No. 169 a on page 114 Paragraph No. 875 now appears as No. 169 h on page 115 Paragraph No. 876 now appears as No. 169 c on page 115 Paragraph No. 377 now appears as No. 226 a on page 154 Paragraph No. 378 now appears as No. 226 h on page 155 Figure No. 108 now appears as No. 37 a on page 114 Figoie No. 109 now appears as No. 61 a on sage 156 THE WATCH ADJUSTER'S MANUAL A Complete and Practical Guide tor Watchmakers in Adjusting; Watches and Chronometers for Isochronism, Position, Heat and Cold. » BY CHARLES EDGAR FRITTS (EXCELBIO Sprina," ■' Electricity ai I This well-known work is now recognized as the standard authority on the adjustments and kindred subjects, both here and in England. It contains an exhaustive consideration of the w various theories proposed, the mechanical principles on which t the adjustments are based, and the different methods followed in actual practice, giving all that is publicly known in the trade, with a large amount of entirely new practical matter not to be ^m found elsewhere, obtained from the best manufacturers and work- ^M men, as well as from the author's own studies and experiences. Sent postpaid to any part of the world on receipt of 8S.60 (los. Sd.) Pnwish^ by THE KEYSTONE THE ORGAN OF TUE JEWELRY AND OPTICAL TKADES 19TH & Brown Sts., Philadelphia, U, S. A. Watch and Clock Escapements A masterly treatise on the lever, cylinder and chro- nometer escapements, with an illustrated history of the evolution and development of the escapement idea in horology. This book is noted for its practical character and for the number and e.\celience of the illustrations. It is the work of two of the most accomplished and experienced teachers of horology in the United States, and their skill in imparting their knowledge to students is shown on everj' page of the book both iti text and in illustration. Though thoroughly scientific, it is lucid in statement and surprisingly easy of comprehension, even to students of limited mathematical or geometrical attainments. Any watchmaker, student of horology or apprentice can easily master it without the aid of an instructor or attendance at a school. It is conceded to be the most complete and lucid work on the three principal escapements available to students of horolc^y. An interesting appendix to the book is an illustrated history of escapements from the first crude conceptions to their present perfection ; also an illustrated article teUing lucidly and step by step how to put in a new cylinder — a most practical bit of in- formation for the «atch repairer. Bound in silk cloth ; 198 pa^es and J16 III astrmt ions. Sent postpaid to any part of the world on receipt of price Sl.&0<'6s.3d.) A:^h^ br The Keystone, ■J OF THE JEWELRY AND OPTICAL TKADBS, ID Brown Sts., Phu-adelphia, U.S.A. THE ART OF ENGRAVING ENGRAVING I A Complete Treatise on the Engra- ver's Art, with Special Reference to Letter and Monogram Engraving. Specially Compiled as a Standard Text-Book for Students and a Reli- able Reference Boolt for Engravers. This work is the only thoroughly reliable and exhaustive treatise published on this important subject. It is an ideal text- book, beginning with the rudiments and leading the student step by step to a complete and practical mastery of tlie art. Back of the authorship is a long experience as a successful engraver, also i a successful career as an instructor in engraving. These qualifica- tions ensure accuracy and reliability of matter, and such a course | of instruction as is best for the learner and quahfied engraver. | The most notable feature of the new treatise is the instruc- tive character of the illustrations. There are over 200 original illustrations by the author. A very complete index facilitates reference to any required topic. Bound In Silk Clotli— 308 Pag'es and 216 lllusIratlODS. Sent postpaid to any part of the worM on receipt of price, SI. 60 16s, 3d.) iHibiisbed by The Keystone, THB ORGAN OP thB jEWHr,iiy and OPTICAI, tradb^ ] 19TH & Brown Sts., Philadelphia, U.S.A. The Keystone Record Book of Watch Repairs Method in repair work makes it essen- tial for every jeweler to keep a specially prepared book for recording watch repairs. The Ideal book of this character is the Keystone Record Book of Watch Repairs, which has space for sixteen hundred entries. Only a scratch of the pen under printed headings and you have a permanent record of watches repaired, charges, etc. This book is necessary for reference, systematizes beach business, and saves labor and time in bookkeeping. It is made of fine, durable paper, has 9 X 1 1 inches, and is substantially bound back and comers. It is much better and one-third lower priced than any other book of its kind on the market. About three-fourths of the jewelry trade now use it. Bound In Cloth, with Leather Back and Corners Sent postpaid to any part of the world on receipt of price, Si.OO (4s. 2d.) pubiiahfd by The Keystone THE ORGAN OF THE JEWELRY AND OPTICAI, TRADSB | 19TH & Brown Sts., Philadelphia, U.S.A. 20 pages measuring 1 cloth, with leather The Keystone Book of Guarantees of Watch Repairs The success of the watch repairer is proportional to HjL Jk \ WjiioiEtEPAKS /■«. the confidence of the public H' 1 Xtnearwmnc/ ''^ ^'^ WOrk. The best workmanship fails from a business stand' point if the workman does not convince his customers that it is the best. The one efiective way to do this is to give a signed guarantee with it, and the cheapest, safest and best form of guarantee for this purpose is found in The Keystone Book of Repair Guarantees, This book contains 200 printed guarantees, and is hand- somely bound. Each guarantee is 2/4 " 7H inches, and most carefully worded. Jewelers have discovered that the t effective way to secure and culti of these guarantees it ; public confidence. Bound In Clatb, wltb Leather Back and Corners Sent postpaid to any part of the world on receipt of price SI. 00 (4s. 2d.) Pubiish..d by The Keystone S ORGAN OP THS JEWEI.EY AND OPTICAI. 19TK & Brown Sts., Philadelphia, U.S.A. THE KEYSTONE BOOK OF MONOGRAMS This book contains 34«> designs and over 6000 different combinations of two and three letters. Is an essentia! to every jeweler's outfiL It is not only necessary for the jeweler's own use and guidance, but also to enable customers to indicate exactly what they want, thus saving time and possible dissatisfaction. The Monograms are purposely left in outline, in order to show clearly how the letters are intertwined or woven together. This permits such enlai^ement or reduction of the Monogram as may be desired, and as much shading, ornamentation and artistic finish as the jeweler may wish to add. This comprehensive compilation of Monograms is especially available as a reference book in busy seasons. Its use saves time, thought and labor, and ensures quick and satisfactory work. Monograms are the fad of the time, and there's money for the jeweler in Monogram engraving. The knowledge in this book can be turned into cash. All the various styles of letters are illustrated. Price, SI.OO (4a. Sd.) bT The Keystone, B ORGAN OF THE JKWEI.RV AND OPTICAl. TRADES, 19TH & Brown Sts., Philadelphia, U. S. A. THE KEYSTONE PORTFOLIO OF MONOGRAMS This portfolio contains 121 com- bination designs. These designs were selected Irom the best of those submitted in a prize competition held by The Keystone, and will be found of value to every one doing engraving. The designs are conceded to be the best in the market, excelling in art and novelty of combination and skill in execution. They are printed from steel plates on stiff, durable paper, and contain sample monograms in a variety of combinatious. The portfolio is a bench require- ment that no jeweler can afford to be ^^ without. It is a necessary supple- ^^L ment to any text-book on letter ^^m engraving. ^o uw Price, 60 Cents (as. pubiish-^j hr The Keystone, ORGAN OF THE JKWEI.RY AND OPTICAI, TRADSS, 19TH & Brown Sts., Philadelphia, U.S.A. THE OPTICIAN'S MANUAL VOL. I. Bv C. H. Brown. M. D. ebJwitiaBia PkiteMpfaU H< if Plulwtcl[4iu Connlf , The Optician's Manual, Vol, I., has proved lo be the most popular work on practical reJraction ever published. The knowledge it contains has been more effective in building up the optical profes- sion than any other educational factor. A study o( it is essential to an intelligent appreciation of Vol. II., for it lays the foundation structure of all optical knowl- edge, as the titles of its ten chapters show : Chapter I.— Introductory Remarks. Chapter II.— The Eye Anatomically. Chapter III.— The Eye Optically; or. The Physiology of Vision. Chapter IV.— Optics. Chipter v.— Lensea. Chapter VI.— Numbering of Lenses. Chapter VII.— The Use and Value of Qlasses. Chapter VII I.— Outfit Required. Chapter IX.-~Method of Examination. Chapter X.— Presbyopia. The Optician's Manual, Vol. I., is complete in itself, and han been the entire optical education of many successful opti- ciaoH, For student and teacher it is the best treatise of its kind, being simple in style, accurate in statement and comprehensive In iU treiitment (if refractive procedure and problems. It merits tllo place of honor beside Vol. II. in every optical library. Bound III Cloiti~433 pag'cs— colored plates and lUustratloos. Mnt postpaid on receipt of SI<BO (6s. 3d.) iMbiiihci by The Kevstone, TKK OHaAH OP THK JEWEIJCY AND OPTICAI, TRADSS, 19TH & Brown Sts.. Philadslphia, U. S, A. THE OPTICIAN'S MANUAL VOL. II. T C. H. Bmmbh. H. O. The Opbdan's Manoal. \'6L IL. s a direct cocitinuation oi The Opbdia'a Mannal, VoL I., btaog a mudi loorc adraaced and comprchensnc treatises It covers in minutest detaO the Iota great subdivisioBS of practical eye rvErac- H y pennctropia. AsttEinatisin. It amtatos the most authoritative and complete researches up to date on these subjects, treated by the master hand ol an eminent oculist and optical teacher. It is thorot^hly prac- tical, explicit in statement and accurate as to fact All lebac- tive errors and complications are clearly explained, and the methods of correction thoroughly elucidated. This book fills the last great want in higher refracti\'e optics, and the knowledge contained in it marks the standard of professionalism. Sonnd In aotli~408 pagrcs— with lllaslratloiis. Sent postpaid on receipt of Sl,60 <6s. 3d.} pabiithed br The Keystone, THB OKCAN op the JEWBtRV ANU OPTICAL TRADSa; igTH & Brown Sts., Philaoblphia, U. S. A. 1 PHYSIOLOGIC OPTICS Ocular Dioptrics— Functions of the Retina— ( novemeots and Binocular Vision AUTHORIZED TRANSLATION Bv CarlWeiland, M.D. Eliilo3tlphta,''pa. ' ' '™"' This translation, now in its second edition, is tlie most masterful treatise on physiologic optics. Its distinguished author is recognized in the world of science as the greatest living authority on this subject, ; his book embodies not only his own researches, but those of several hundred investigators, who, in the past hundred years, made the eye their | specialty and life study. Tscheming has sifted the gold of all optical research from the dross, and his bool<, as revised and enlarged by himself for the purposes of this translation. Is the most valuable mine of reliable optical knowledge within the reach of ophtliatmologists. It contains 380 pages and 212 Illustra- tions, and its reference list comprises the entire galaxy of scientists who devoted their researches to this subject. The chapters on Ophthalmometry, Ophthalmoscopy, Accommodation, Astigmatism, At>erraHon and Entopic Phenomena, etc — in fact, the entire t\v>k contains so much that is new, practical and necessary ttait no wltactionist can afford tw be without it, Konnd In Cloih. 3S0 pj^es, 3i3 tlliuintloiis Prt«. S2.BO iios.tM.> r.««**a^5THE KeVSTONE i«TM A Rrohcn Srs., pRnjiocunnA, U-S.A. THE PRINCIPLES OF REFRACTION in the Human Eye, Based on the Laws of Conjugate Foci Bv Swan M. Burnett. M. D., Ph. D. PrOfMlor of Opbthalmoil lolOBv In tl ■r Cilaic, t In this treatise the student Is given a condensed but thor- ough grounding in tlie principles of refraction according to a method which is both easy and fundamental. The few laws governing the conjugate foci lie at the basis of whatever pertains to the relations of the object and its image. To bring all the phenomena manifest in the refraction of the human eye consecutively under a common explanation by these simple laws is, we believe, here undertaken for the first time. ■ The comprehension of much which has hitherto seemed difficult to the average student has thus been rendered much easier. This is especially true of the theory of Skiascopy, which is here eluci- dated in a manner much more simple and direct than by any Imethod hitherto offered. The authorship is sufficient assurance of the thoroughness of the work. Dr. Burnett is recognized as one of the greatest authorities on eye refraction, and this treatise may be described as the crystallization of his life-work in this field, execu ma the The text is elucidated by 24 original diagrams, which were executed by Chas, F. Prentice, M. E. , whose pre-eminence in mathematical optics is recognized by all ophthalmologists. Sent postpaid t< Bound in SMk Cloth. an; part of tbe world on receipt of price, SI.OO (4s. 3d.) Published by The Keystone, THE ORGAN OP THE JEWElvRY AND OPTICAl. J9TK AND Brown Sts., Philadelphia, U.S.A. SKIASCOPY AND THE USE OF THE RETINOSCOPE A Treatise on the Shadow Test in its Practical Application to the Work of Refraction, with an Ex- planation in Detail of the Optical Principles on which the Science is Based. This new work, the sale of which has already necessitated a second edition, far excels all previous treatises on the subject in comprehensiveness and practical value to the refractionJst It not only explains the test, but expounds iuUy and explicidy the principles underlying it — not only the phenomena revealed by the test, but the why and wherefore of such phenomena. It contains a full description of skiascopic apparatus, including the latest and most approved instruments. In depth of research, wealth of illustration and scientific completeness this work is unique. Bound In cloth; contains 331 pases and 73 lUustrationa and colored plates. Sent posipald la anj part of the world on receipt of SI .OO (4s. 2d.) PuMi^hfdby The Keystone, VN OP THB jaWHLRY AND OPTlCAi TRADBS. AND Brown Sts,, Philadelphia, U.S.A. OPHTHALMIC LENSES Dioptric FormulsB for Combined Cylindrical Lenses, me Prism-DiOptry and other original Papers Bi Charles F. Prentice, M. C. A new and revised editioD of all the original papers of this noted author, combined in one volume. In this revised form, with the addition of recent research, these standard papers are of increased value. Com- Uned for the first lime in one volume, they are the greatest compilation on the subject of lenses extant. This book of over 200 pages contains the following papers : The PriBm-Dlopi tar Comlriaed CyliadHcffl Lemeit il Nuraberfais and Measorlnfc Prim The PrUm On tile Practical Etecullon ol Opbthallnic PreKriptlons invDlvlns Prinni. A Problem Id Cemented Bifocal Lenaei. Solved by tbe Pri^m-Dioplry. Why Stmne Conlra-aeneric Lense* of Equal Power Pall to Neutralize Correction of Depleted DyDamlc Refraction (Presbyopia) Press Notices on the OrlglnaJ Edition: OPHTHALMIC LENSES. AiRv England AfediceJ Geietti. ' kkA on ihii ipeclil lubjccl ev< ■—Harological Rnicw, New DIOPTRIC FORMULA F CYLINDRICAL LENSES. of ophilmJ- iDgianilaicd I Bi laban. 1 w >iinple "'It- ■led by H. Knapp, The book contains HO Original DiagramB. Bound In cloth. Price, $l.50 (6s. 3d.) Published by The Keystone, THE ORGAN OP THE JEWELRY AND OPTICA: 19TH St Bkown Sts,, Philadelphia, U. S. A. 3 TESTS AND STUDIES OF THE OCULAR MUSCLES Bv Erncst E. Madoox, M. D., F. R. C. S., Ed. t)phlluliDls •uiRogn lu \he Uaytl VlclorU Boapll^ BoDmemoaih ; torroer); aai^ ophlhjdnilt' qurgttiD lo the Roysl loDniuiry, Ediobur^h SECOND EDITION 4>rt«ed and enlurg^ bj the autb The suli-division of rpfractive work that most troubles the optician is muscular anomaiies. Even those who have mastered all the other intricacies of visual correction will often find their Hkill frustrated and their efTorts nulUfied if they have not thoroughly mastered the ocular muscles— their functions and failings. The optician can thoroughly equip himself in this fundamental essential by studying the excellent treatise "The Ocular Muscles," by Dr. Maddox, who is recognized in the medical world as one of the leading authorities on the subject ol eye muscles. This work is the most complete and masterful ever com- piled on this important branch of ophthalmology, covering thoroughly the symptoms, tests and treatment of muscular anomalies. The accomplished author has devoted a lifetime of study and research to his subject, and the book throughout is marked by an explicitness and simphcity of language that make its study a pleasure to the eye specialist. Bound In Clotli. Over too Well Executed Illustrations Sent postpaid to an; part of the vorld on receipt of SI. 60 (6s. 3d.) PQbiish^ hj The Keystone THE OKGAN OF THK JEWELRY AHO OPTICil. TRADES 19TH & Brown Sts., Philadelphia, U. S. A. Optometric Record Book A record book, wherein to record optometric examinaitions. is an indispensable adjunct of an optician's outfit The Keystone Optometric Record Book was specially pre* pared for this purpose. It excels all others in being not only a record book, but an invaluable guide in examination. The book contains two hundred record forms with printed headings, suggesting, in the proper order, the course of examina* tion that should be pursued to obtain most accurate results. Each book has an index, which enables the optician to refer instandy to the case of any particular patient The Keystone Record Book diminishes the time and labor required for examinations, obviates possible oversights from carelessness and assures a systematic and thorough examination of the eye, as well as furnishes a permanent record of all exam* inations. Sent postpaid oo receipt off SI >00 (48. 2d.) Published by The Keystone, THE ORGAN OP THE JEWELRY AND OPTICAI, TRADBSS 19TH & Brown Sts., Philadelphia, U.S.A. VZ3U.